,tOMATHEMATICS TRf\IN\NG PROGtWl.
t This work was partially supported by NSF Grant GP-42325.
*Work done while author was on leave from
Universit~
de Strasbourg, France.
BlOlM1l\EtAAllcs
...
)-
STUDIES IN ENUMERATION
t
Dominique Foata*
Department of Statistics
University of North Carolina, Chapel Hill
Institute of Statistios Mimeo Series No. 974
January, 1975
lRMN\NG PROGRMI
STUDIES IN ENill1ERATION
by Dominique Foata
Notes of a seminar held at the department of Statistics of the Univversity of North Carolina, Chapel Hill, during Fall semester 1974.
-2-
TABLE OF CONTENTS
INTRODUCTION
1•
...
REARRAr~GEHENTS
3
OF SEQUENCES
9
1.
The fundamental transformation.
9
2.
Eulerian numbers
16
3.
Slicing the cube.
22
4.
The fundamental transformation in the multipermutation
case
27
5.
Recapitulation of the previous section and intercalation
monoids
43
6.
The fundamental transformation of the symmetric semigroup 57
7.
A probabilistic lemma and some applications
8.
The equivalence principle of Sparre Andersen.
braic study.
9.
Bohnenblust rearrangement theorem
75
An agle-
84
95
APPENDIX
109
BIBLIOGRAPHY
113
-3-
INTRODUCTION
It may seem surprising to write seminar Notes under a single chapter,
especially since the above title suggests a broad coverage of various results
on Enumeration.
It was indeed the author's intention to discuss at least
four aspects of this topic.
Four chapters were originally scheduled, but
only the first one was written and is the content of these Notes.
It
seems appropriate to mention the organization of the other three chapters,
which may be reproduced in a further edition of the Notes.
Chapter 2:
Enumeration of permutations.
- Statistics defined on the symmetric group (descents, rises, peaks,
upper records).
- Codings of permutations.
- Inversion and major indices.
- Explicit and recurrence formulas for counting permutations.
Chapter 3:
Combinatorial analysis of numbers.
- Geometric studies of Bernoulli, Euler and Genocchi numbers.
Chapter 4:
Connection between symmetric group representation and enumeration.
- Young tableaux, SChur functions, Robinson-Schensted - SchUtzenberger
.~
constructions, Foulkes formula for the number of permutations with a prescribed up-down sequence.
-4-
During the Fall semester 1974 chapters 1 and 4 were presented in
the seminar, but only chapter 1 was fully written.
The appendix at the
end of the Notes contains however an annotated bibliography about the
content of chapter 4.
When enumerating permutations according to certain patterns (such as,
with given maxima and minima, with a fixed up-down sequence, ..• ) one is
often led to transfer the counting problem to another class of permutations for which the problem is straightforward, or at least easier.
Of
course, there is no general rule, but we have at our disposal several
natural algorithms which make this transfer possible.
The purpose of
chapter 1 is to give the description of those algorithms and mention
several applications.
Had we covered the content of chapter 3 on Com-
binatorial Theory of Numbers, it would have been clear that those algorithms
are basic when combinatorial interpretations of classical sequences of
numbers are sought.
is the following:
The typical set-up for describing those algorithms
let
S
xl x 2 ••• x whose elen
U and V be two real-valued
be a set of sequences
ments belong to a totally ordered set; let
functions defined on
S.
Then, a bijection
of
~
S onto itself has
xl x2 ... xn is
mapped onto a rearrangement of its terms, say, X. x. • •• x. , where
11
12
In
. .. i
is a permutation of 1 2 .•. n
with the subsidiary proto be constructed in such a way that the sequence
n
.. .
...
x. ) = U(x l x
x ) . Most of the time the
2
n
In
2
definition of the bijection ~ is of algorithmic nature, that is to
perty that
Vex.
11
x.
1
say, no "explicit" formula exists for
~(xl
x2
rearrangement has to be defined step by step.
...
x ) ,
n
bur rather the
· -5-
The first example of such a rearrangement is described in section 1.
It is the so-called "fundamental transformation".
It is
a bijection of the permutation group of the set of n(n > 0)
elements onto itself.
Its construction is based upon the fact
that any permutation of 1 2 •.. n can be expressed either as a
word
with x.J.
tation
the image of
i
under the permu-
(1 s i s n),
or as a product of disjoint cycles written
in a suitable manner.
It is worth noting that such a simple con-
struction already gives non-trivial results about the distributions of several statistics defined on the permutation group
S
In particular, the generating function of either the
n
number of descents, or the number of so-called exccedances over
S
n
is the celebrated Eulerian polynomial
2).
A (t)
n
(see section
In section 3 we mention two problems on Eulerian numbers,
whose solutions are known but remain unsatisfactory.
In section 4 the fundamental transformation is extended to any
set of sequences having repetitions.
There is some amount of
algebraic work to do in order to achieve that extension.
In
particular, a substitute for the notion of cycle, which the
fundamental transformation of the permutation group was based
upon,
has to be found.
The algebraic structure to be intro-
duced is the airauit monoid,
the free monoid
which is a quotient monoid of
x* derived by a set of commutation rules.
-6-
The intercalation monoid, defined in section 5, is isomorphic
to the circuit monoid.
generalization of
It is used here to obtain a non-commutative
~1acMahon's
"Master Theorem".
Section 6 contains another extension of the fundamental transformation to the set of all maps of the interval
into itself.
Several applications are given.
[n]
= {I,
2, ••• , n}
For instance, the
Abel identities can be established by interpreting left and right
members in a suitable manner.
also provides
with~cl0sed
The fundamental transformation
formula for the bivariate generating
function of the number of fixed points and number of one's over
the set of acyclic maps.
Section 7 contains a lemma that makes possible the transcription of theorems on rearrangements into probabilistic
langu~ge.
An algebraic study of the so-called equivalence principle of Sparre
Andersen is the content of section
8.
Finally, in section 9
Bohenblust rearrangement theorem is presented.
Let us recapitulate the rearrangement algorithms that will be
found in the present Notes:
- the fundamental transformation of the symmetric group (section 1);
-7-
- the fundamental transformation of sets of multiperrnutations
(section 4);
- the fundamental transformation of the symmetric semi-group (section 6);
- Ian Richards' map (section 8; lemma 36);
- Bohnenblust transformation (section 9).
-8-
ACKNOMLEDGH1ENTS
I am very grateful to the colleagues and friends who attended
the seminar and whose remarks unwittingly,' influenced portions of
the Notes.
In particular, I thank Robert
Davi~
Ulrich Faigl, Ladnor
Geissinger, William Graves, Douglas Kelly, and Dennis Kinch.
The
critical remarks of Rhodes Peel have also been of great value in
the preparation of the Notes.
Finally, I am most grateful to
Indra Chakravarti who was at the origin of the organization of
the seminar.
He was kind enough to discuss with 'me the ingredients
and make several suggestions.
The typing of the Notes was due to Jackie O'Neal, of the department
of Statistics.
I thank her for her skillful work.
-9-
1.
The FunJamental Trans'formation.
The so-called Yifundamental transforr.lation H , hereafter denoted by
a -+
8,
of the permutation group appears to be the keystone in a great
nlany combinatorial constructions.
D and
tnat
E that will be shown to be identically distributed by observing
Ea
Let
= D8
a = xl
a word of the
x
n
2
... x
n
letters
lsi < n
and
n
a,
=3
i
written as
1 2 ... n
in some order.
We denote by
i.e. the number of integers
i
1 s i < nand
such that
the values taken by
D and
x.
1
>
i
such that
is denoted by
E appear in the following
table
a
We
observe
that for
same distribution over
n
S
n
Da
Ea
123
0
0
132
1
1
213
1
1
231
1
2
312
1
1
321
2
1
=3
the two statistics
D and
TIle same result holds for any
be further shown in this section.
Da
i.e. the
number of exceedances of a
The
number of integers
For
be a permutation of
1, 2, ... , n
the number of descents of
,ca.
First let us introduce two statistics
E have the
n
as will
-10-
Let
distinct
w= x x
1
2
...
x be a word of m letters, not necessarily
m
The first letter xl is denoted by Fw = x
The
1
.
.
(m > 0)
word w' is saiJ. to be initiaZZy dominated if xl > x.
~
such that
1 < i :::; m
.
holds for all
How an inareasing faatorization of w is a sequence
of initially dominated words such that
(WI' w2 ' ... , wp )
i
position product of
W
is the juxta-
Le.
=
w
and
Fw
P
For instance the words
w
= 38417926
and
Wi
= 3122315462336
admit the
increasing factorizations
(3, 84517, 926)
and
(3122, 31, 54, 6233, 6) ,
respectively.
1.1. Lemma. Any word w = Xl
x
2
••.
X
admits one and onZy one inareasing
m
f aatonzation.
Proof.
x.~
We say that the letter
1 < i :::; .n and
x. :::; x.
J
~
for all
is outstanding in
j < i
When
.
n
w
1'.c
....
i
=1
cutting i1 the word
or
w just
before each outstanding letter we clearly obtain an increasing factorization.
It remains to show that it is the only one.
Suppose that there are two such factorizations, say
and
..., ws )
We can assume that
v.
J
Let
j
be the smallest index such that
is shorter than
w. .
J
... ,
v )
v.
w.•
J
;t
r
J
for
-11-
some non-empty word u
Fw. > Fu
we h,ave
J
anci
= FV j + l
Fu = Fv. I
J+
Fw.
J
= Fv.J
First, let
=
T
me·j··~ax
. I).
T
(J
wilen
-+ IJ
m-I ("i'laX
•
I) •..
rearrangement of the m elements of
T
dominated rearrangements of the word
x
x
1
x
2
< ;:iax I
(1)
T
2
, ... ,
T
r
be the restrictions of
they are all cyclic permutations, we can form
We then let
IJ
of IJ.
hI' x2 '
••• , x }
m
Clearly q
is a
is a
onto the set of initially
m
r
orbits
II' 1 , •.. , I r
2
r
to
(J
As
q(T )
I
be equal to the juxtaposition product
=
1be sequence
I =
a way that
nW1bered in such
,
is a p erJ:lUtat ion .
Otlax I) .
Next take a permutation of a finite set having
l
(J
in some order.
I
I
T
is
leading to a con-
Furthermore q(T)
bijection of the set of cyclic permutations of
Let
vr )
Q.E.D.
'tm· e""
iftax I) =i1axI.
is cyclic, we have
T
J+
... ,
We put
q (T)
As
Fv. I '
T be a cyclic permutation of a finite set
integers.
ill
~
ev l , v 2 '
TIms the factorization is unique.
Let us now give the construction of
of
is initially dominated,
J
On the other hanel, as
an increasing factorization, we get
tradiction.
w.
As
q(T ) .
r
eq(T ), q('t )' ••• , q(T )) is precisely the increasing sequence
Z
I
r
The mapping (J -+ IJ is bijective, since there corresponds to (J one
-12-
and only one sequence of cyclic permutations of sets
union
[n]
such that
(1)
holds.
II' 1 , ... , I r
2
To such a sequence
(L , L , ...• Lr)there
2
I
w ) of initially
corresponds next one and only one sequence
dominated words such that
rearrangement of
FW
1
1 2 •.. n.
(WI' w2 ' .•. , wr )
of
r
and WI w ... wr be a
2
r
From lemma 1 there finally corresponds to
< FW
2
< ••• < FW
~
one and only one permutation
admitting
(WI' w2 ' ... , wr )
as its increasing factorization.
2.
Example.
Consider the permutation
a
=
[:
2
3
4
5
6
7
8
3
6
9
5
1
4
2
:] .
The orbits written in increasing order according to their maxima are
11 = {5} , 1 = {l, 3, 6} , IS = {2,8} , I 4 = {4, 7, 9} •
2
Let
Lo
be the restriction of
J
a
to
I. (1
J
s
j
s 4)
.
Then
q(L 1 ) = 5
q(L 2 ) =
1:~(6)
2
1: (6) 1: 2 (6) = 631
2
2
q(1: ) = L (8) 1: (8)
3
3
S
q(1: )
4
Hence
~
= 563182947
(5, 631, 82, 947) .
= 1"43 (9)
2
1: (9)
4
= 82
'r
4
(9} = 947 .
Note that the increasing factorization of
~
is
-13-
In the general case to construct
e,
increasing factorization of
tion
a
a
say
e we have
w , ... , w ) .
z
r
from
(wI'
is then the product of the disjoint cycles
be a \vord.
Let
We denote by w
its non-decreasing rearrangement is
letters
n
ment of w is then
gers.
and
over,
m
l
l TIl
+
m
Z
+ ••• +
m
2 2 ... n n
1
-1
(wZ)
1
... q (wr )
= xl
itself.
If the
letters 1, m letters
l
Z
the non-decreasing rearrange-
ill
m = m),
n
Let (x, y)
be an ordered pair of inte-
We denote by v
(w) the number of integers i such that 1 ~ i ~ m
x.y
xi = x • xi = y. In the same manner ~x,y(w) denotes the number of
integers
and
n (m
(wI) q
1 2 ••• n
letters of ware not distinct, containing (say)
2, ... , m
-1
The permuta-
x 2 ••• xm its
For instance, if w is a permutation of
non-decreasing rearrangement.
1 2 ... n,
q
to take the
i
~x,y(w)
\'Ie
such that
1
i < m and
~
can only be equal to
0
x.1
or
=y
1
,
xi + l
=x
.
Clearly
if w is a permutation.
v
(w)
x,y
More-
have
Dw
(2)
I
=
x<y
~x,y(w)
and
Ew
=
I
x<y
3.
.
Proposition.
For any pair of integers
a.
"x,y(w)
(x,y)
such that
we have
=
~x,y(e) .
x < y and any permutation
-14-
Proof.
Let
for SOtle
1'.
]
i
a
.
= xl
x2
But
i
x
n
and x.
If
be the restriction of a to
x.i
= xi
x
2 ...
Conversely, let
e
factor
of the word
J
x~
x!1 x!1+ 1
have
4.
y
~o~d
w.
J
yx
=i
x. = y
1
<
.
1.
J
in this order and so
Let
q('"C. )
]
~x,y(e) = 1
(e) = 1, there is one
x,y
that is equal to yx. Let (WI' wz'
If
.
be the increasing sequence of e
the last letter of a
x
Then, the dominated work
1.
that is
1
then
belong to the same orbit, say
1
contains the letters
=1
Yx,y(a)
As
~
the letter
x!1 > x!1+ 1 '
x!1
... ,
Hence
J+
= yx occurs inside some word w.]
= '"C.(x),
hence y = a(x), i.e.
]
With
v
x,y
(a)
q -1 (w . )
=
1'.
J
=1
J
we then
.
Q.E.D.
Corollary.
For any pemrutation
a,
we have
= De
Ea
Proof.
TIle corollary follows in1mediately from the previous lemma and
Q.E.D.
formulas (2) and (3) .
Other statistics have the same distribution as
be stated without proof.
((1970), Chap. 1).
rises, denoted by
and
wp )
cannot be
x!1+ 1 the first letter of w. 1
and
.
x. < x. 1
]+
]
(by convention x
0
.
a
= Xl
x
x
2
is the number of integers j
r - exoeedanaes, denoted by Er a
and x. ~ j + r
Of course E
J
E, as will now
Further details can be found in Foata-SchUtzenberger
For any permutation
Ra,
D and
1
= 0) .
For any r
is the number of
=E
j
n
the number of
such that
~
0
~
j < n
0 the number of
such that
1
~
j
~
n
-15-
5.
Proposition.
The statistics
1 + E)
over Sn
Proof.
vtlhen
Eo
and M have the same distribution as
let
(the reversed image).
Clearly
a
a
a
1
= x
x
2
2
= 6'1
3
= 0"2
a denote the permutation a
+
Da
= Ma.
3
...
xn Xl
1
1 + D (or
= xn •••
Next let
.
TIlen it can be verified that
Q.E.D.
6.
Example •..
Let
a
=
(1 2 3 4 5 6 7)
ll16!2. 25 ·
a
a
1
2
a...,)
=
r
2 3 4 5 6 7]
1647253
=
1 6 2 5 7 4 3
=
3 4 7 5 261
MaS . . = 4 •
-:16-
2.
Eulerian
~umbers.
For
on n.
having
and
k descents.
insert
n
A
let
k < n
~
0
n,k
denote the number of permutations in Sn
Take a permutation
just
is readily obtained by induction
E over Sn
The distribution of D or
after
x
k
(1
~
a
~
k
= Xl Xz
n - 1)
X _ (n > 1)
n 1
or
before
x > x +1 ' or k = n-1» the number of
k
k
remains alike. On the contrary, when 1 ~ k < n - 1 and xk <
when n is placed just before Xl' the number of descents is
If
1
~
k < n - 1 and
o~
k
~
=1
He have then A1 ,O
by one.
, A1 ,k
=0
for
k
~
X
descents
xk+ l '
or
increased
1 and fer n
~
2 and
n - 1
A
n,k
=
(k
+
1) An - 1 ,k
(n - k) An - 1 ,k-1 .
+
The above formula is the well-known recurrence relation for the Eulerian
nwnbers.
TIle first values are shown in table 1
k
A
n
.
n,K
1
0
4
1
1
1
1
5
6
1
1
1
2
3
2
3
4
5
1
26
30Z
1
57
1
1
1
11
4
11
66
302
26
57
TABLE 1
With
t
an indeterminate the polynomial
An (t)
=
I
A
O~k~n-1 n,k
t
is called the Eulerian polynomial of order
k
n .
It is the generating
-17-
polynomial of D (or
over S
E)
=
A (t)
n
Let S
(0
~
n,m
be the set of permutations xl x
m ~ n - 1).
permutation
an
2
• •. xn
such that
We can form such a permutation by taking a subset . I
of cardinality
[n - 1],
namely
n
m,
of the set
permutation groups of
J
and
I
then a permutation
a'
of
= [n - 1]\1. Denoting by
J
we get for
I,
SI
of
finally a
and SJ
the
0 $ m< n - 1
Am(t) tAn- 1 -m (t),
since the definition of a descent depends only on the mutual order of the
Da
elements. Also L{t : a € Sn,n- I} = An- let) . Altogethe~with AO(t) = 1 ,
(6)
A (t)
n
=A
n-
let)
+
t
L(n-l] Am(t) An_l_m(t) (n > 0) •
O$m<n-l m
But it is known (see e.g. Foata (1974) p. 18) that if two sequences
and
(bn ) n~l
of elements of a ring
(a)
n n~l
n are opiven. the following two identities
are equivalent
(7)
a
n
= bn
L
+ l$i$n.. l
(n-lJ a. b .
i
:l
n-l.
(n > 0)
and
1 +
Lan
n>O
un/n 1
= exp( l
.
b unln I) •
n>O n
The latter relation is an identity in the algebra of power series in one
variable
u with coefficients in the ring
Q.
With the change of variables
-18-
i
=n
=I
- 1 - m (6) is rewritten Al (t)
and for
A.1 (t)(tAn- I -1. (t))
n > I
+
[n-1]
n-l
An- let)
.
1 •
Comparing with (7) this lead to
(9)
1 +
l
n>O
A (t) un/nl
n
= exp(u
+
l
n>l
tA let) un/n!) •
n-
It is an impZicit relation for the exponential generating function of the
In order to obtain an expZicit formula we can make
Eulerian polynomials.
use of the following argument.
From proposition 5 it follows that the genera-
ting polynomial of the number of rises over S
n
l{t
RO
:
°€
Sn }
= tAn (t)
is
(n > 0) •
With the same notations as above we then have
tA (t) • tA 1 (t)
m
n- -m
and this time the relation holds for
vention on counting rises.
tAn (t)
0
~
m ~ n - I,
because of our con-
Hence
=
(n > 0)
or
(n > 0) ,
which is equivalent to
(10)
1 +
l
n>O
tA (t) un/n!
n
= exp l
tA let) un/n! .
n>O n-
-19-
Let
D(u)
=1
I A (t) un/nl
n>O n
+
=1
R(u)
(11)
and
+
R(u)
=1 +
L tA (t) ull/nl .
n> O n
Then
t(D(u) - 1) •
From (9) we deduce
exp
L tAn_let)
= exp(-u) D(u) ,
un/nl
n>l
and from (10)
= exp(ut
R(u)
- u) D(u) .
The system of the two equations (11) and (12) readily gives
=1
D(u)
(13)
+
L A (t) un/n!
n>O n
= (1
-
L tAn(t)
= (1
- t)/(l - t exp(u - ut)) .
t)/(- t + exp(ut - u))
and
(14)
R(u)
=1
+
~n/n!
n>O
Ibe next formulas about Eulerian numbers and polynomials will be given
without proof.
(1970).
For further details see Carlitz (1958) or Foata-SchUtzenberger
For m > 0 and n
n
m
(15)
~
=
0
we have
I
0~k~n-1
(m+k) A k .
n
n,
Actually the preceding ;iCentity can' be establishec comlJinatorially by interpreting in a suitable manner the two members.
1/(1 _ t)n+1
=
Using the identity
L.t k [n+k) ,
O~k
n
it is easy to see that (15) implies the so-called Worpitzky formula
(16)
A
n,k
= L
(-1) i (k
O~i~k
(0 ::; k
~ II -
1
n > 0) .
-20-
We conclude this section by giving a connection between EuZerian and
EuZer numbers.
With the change of variables
complex number of modulus 1 and argument
1
Denoting by
+
n/2)
t
-+ -
(D 2p - I )p>0
(i
the
formua1 (13) becomes
the sequence of tangent numbers, namely
L D2
p>O
-1 u
2p l
- /(2p_l)!
P
p > 0 we have
A
2p (-1)
(17)
(-1)
p-I
=0
_
Azp_l(-l) - D2p _1 .
These two relations can be proved combinatorially
counts the permutations
xl
alternating i.e. such that
••• J
iu
-+
A (-1) (iu)n/n! :: 2/(1 + e- 2iu ) :: tan u .
n>O n
we conclude that for any
i :: 1,
u
I
tan u::
any
1 and
z ...
r~nembering
_ of 1 2 ... (2p-l)
Zp 1
is less than bot~1 x,).
1 and
,;.1-
X
D
2p-l
that are
for
2i
(see e.g. Foata-SchUtzenberger (1970), Chap. 5).
p - 1
x
x
that
We also know since Desire Andre (1381) that the number of alternating
...
permutations
...
x2p of 1 2
2p (same definition as above with
xl x2
the extra condition: x
> X ) is equal to the coefficient D
of
2p
Zp
2p-I
Zp
u / (2p) ! in the development of sec u :
sec u :: 1
The coefficients
+
D u 2p /(2p)!
p>O ZP
l
(° 2p ) p> 0 are called secant (or EuZer) numbers.
kind of relations as (17) hold for secant nmnbers.
Roselle (1968).
Let
3
n
be the set of permutations
The same
They were found by
xl
z ...
X
x
n
of
---------------
------~
-21that are without suaaessions, which means that
1 2 .•• n
1
+
x.1
X.
;Ie
1+
1
for all
i = 1, 2, ... , n - 1.
1
x
;Ie
1
and
TIlen Roselle (1963) intro-
duced the polynomials
B (t)
n
Thus
.-;.~
iJ
n
B
n
(t)
= l{t Rcr :
cr
€
B }
n
(n > 0) .
is the generating function of the number of rises over the set
Let
(n > 0) .
The first values of the coefficients
,k
1
2
B
n,k
appear in table 2.
3
4
S
6
1
21
161
813
1
51
813
1
113
1
n4---::----~---...::-----..:..-----:;.----....;;;.....
l
0
2
1
3
4
5
1
1
1
6
1
21
Sl
7
1
113
1
7
TABLE 2
Tnen
Roselle (1963) established the identities
B _ C-l)
2p 1
(18)
(-l)P B2p C-1)
= D2p
=0
(p >
0) .
In Foata-SchUtzenberger (1970), Chap. S, a combinatorial proof of these two
identities is also given.
In particular, it is shown that
B (t)
n
is also
the generating polynomial of the number of O-exceedances over the set
of the permuations
xl x
2
•.• x without fixed points
n
(x.
1
;Ie
i
vn
for all
i) .
-22Thus
B (t) = lit
n
51 icing
3.
faa
: a
€
Vn }
t:l-:: Cube.
As we now see, relation (16) also occurs in an entirely different context.
Let
Xl' X2 , •.. , Xn be n (n > 0) independent random variables distributed uniformly over [0,1]:. Let S = Xl + X + ••• + X • The distribution of
n
n
2
Sn is well-known.
It was already obtained by Laplace (1820), but also redis-
covered periodically at least by non-probabilists.
x+
= max(x,O)
7.
Theorem.
(See e.g. Feller, vol. 2 (1966), p. 27).
n
In particular, when
x}
x
= (l/nl) I (_l)i[~](X
O<'~
.-;~",n
~
n
= (l/nl) I (_l)i[~)(k
O:;;;i<k
~
- i)n .
1
0:;;; k < n
P{k < S :;;; k
n
+
I} = (lin I)
I I (-1 / [~]1 (k
I ~:;;;i<k+l
- I (_l)i[~)(k
O:;;;i<k
Hence
P{k
<osn :; ; k
+
I}
= (l/nl)
I
O:;;;i:;;;k
(-l)\k
x
- . )n+
is equal to an integer k of the interval
PiS :;;; k}
for
let
For any real,
we obtain
TIlUS
x
•
PiS :;;;
(19)
For any real
~
+ 1 - i) n
- i)lll.
J
[O,n]
-23-
Now, comparing with (16) we conclude that
P{k < Sn
(20)
~
k
+
l}
= An, kin!
(0
~
k
~
n - 1) •
So the following question arises: aan (20) be direatZy
referenae to formuZas
(16) and (19)7
proved~
without
Yl , YZ' ... , Yn be n mutually
independent identically distributed random variables having an absolutely continuous distribution (in order to have
the form
{Yo
1
=
(i
Y.}
J
0 for the probability of events of
D, the number of desaents
We can define
j)) .
;t
Let
(Y , Y ' ... , Yn ) , as being the number of i such that
l
Z
Y.1 > Y.1+1. We already know the distribution of D, namely
within the sample
1
~
i < nand
P{D
= k} = An, kin!
,
(0
~
k
~
n - 1) •
Moreover,that distribution remains the same for all samples
having an absolutely continuous distribution.
(Y , Y2 , ... , Yn )
l
Accordingly, we should be able
to establish the identity
P{k < S
(21)
n
~
k
+
I}
= P{D = k}
(0
~
k
~
n - 1)
directly.
Geometrically the left member of (21) is the volumne of the subset of
the unit cube in n
=k
and
dimensions comprised between the planes
+ x
set of all points
... , x )
n
n
=k
+
I
Xl + x
+
+ X
2
n
The right member is the volume of the
(xl' x2 ' ..• , xn ) of the unit cube such that the sequence
has exactly k descents.
The second open question about Eulerian numbers, or rather a question
that would deserve a better proof occurs in sorting problems.
A run in a
-24-
permutation
p
(0 < p
(i)
~
xl x2 •.. xn is a factor xk xk +l .•• xk+p _1 ' of length
n), such that the following three conditions hold
either
k
=I
<
(iii)
1
or
either k + P - 1
k
~
nand
x _l
k
>
x ;
k
x
k+p-l
=n
For instance, the permutation
56, 1349, 8, 27.
<
xk+p _1 > xk+p •
a = 5 6 1 3 4 9 8 2 7 has four runs
or k + P - 1 < nand
Clearly, the number of runs in a permutation is equal
to the number of descents plus one.
also counts the permutations of
Consequently, the coefficient
I 2 ... n with
(k
+
1)
runs.
evaluate the avepage Zength of the k-th pun in a permutation.
of course a further definition.
1 2 ..• n
(n > 0)
having
Let
a
= Xl
i O = 0 and
run of a
i d+ l
=n
n,k
Let us
Tais requires
be a permutation of
d descents, say
...
,
Put
x2 •.. xn
A
For any k
S
d
,
+
1 the length of the
k-th
is defined by
(22)
For k > d + 1 we let
For instance, for the above permutation a
=5
6 1 3 4 9 8 2 7 we have
-25-
Sn the problem is to calculate
Assuming that tnere is equirepartition upon
the expected value
o<
and
WI +
k
EU k of Uk(k
0)
.
For any permutation
i s n define the random variable
...
+
Uk
~
m}
(0 < k, m s n)
runs in the sequence
at most
>
k
runs in
(nm]
(lIn!) •
I
'<k
A
by
1
= xl
X. (a) = x.
1
1
x2
...
The event
occurs if and only if there are at most
X)
(Xl'
But the probability of having
m
... , xm)
(Xl' X2 ,
1-
X.
a
. 1 • (n -
ffi,l-
is equal to
m) I
= (l/ml)
I
A
••
Osisk-1 m,l
Thus
=
L P{Ui
As
(l/ml)
I A ..
0::>isk-1 m,l
we._conc1ude that
+
m>O
I
(23)
O<mSn
To indicate the dependence on n
(l/ml) A k-l
- m,
let us write· u~n)
instead of Uk'
Then
we have the following result (see Knuth, vol. 3 (1973), pp. 40-43)
lim
(24)
lim
k
n
EU(n)
k
=
2
or
limk
In other words, for large n
2
when k beoomes 'large.
only proof
!~:iOW~'+
I
O<m
(l/m!) A ' l
m,K-
=2
•
the average 'length of the k-th run tends to
A probabi'listio proof wou'ld be we'loome.
sq{ar
app~als
to .complex variable techniques,
L tku
= t(l -
t)/(- t + exp(t - 1)) -
the expression of
(25)
k>O
k
t
,
TIle
sta~ting
with
xn
-26-
where
t
is a complex variable and
(26)
Uk
= O<m
I (l/mI)
A
1
m, (
(k > 0) ,
and studying the behavior of the coefficient
uk
for large
k.
The
explicit formula (25) is obtained from (13) by making the substitution
u
L A kin I is convergent for all k) .
n>O n, '
. Let us conclude this section by mentioning two results of Carlitz et al
=1
(and knowing that the series
(1972) on the asymptotic behavior of A k'
l1ith x a real and
11,
X
n
= ((n
+ 1)1
l2)1/2 x
(n - 1)/2
+
for any n
>
0
then
lim
(27)
n
and
(28)
= (6/(w(n
(l/nI) An, [x ]
n
+
1)))
1/2
e-
x 2/2
+
. -3/4
Oen
).
In order to prove (27) Carlitz et al (1972) introduced a sequence of ranX k (1 $ k ~ n) that are mutually independent by rows but
n,
not identically distributed and such that the sum
dom variables
sn
=
L X
l$k$n n,k
(n > 0)
has the distribution
P{S
n =
k} =
An,k_l /nI
(1
~
k $ n)
.
-27-
4.
The Fundamental Transformation in the Rultipermutation Case.
Given n
>
0 and n positive integers
with sum m
let us consider the set
S(ml' m , ••• , m ) of all rearrangements
n m
2
m
m
2
l
n
xl x 2 ••• xm of the sequence 1 2
••• n
(as it is well-known the
number of elements of
m I))
It
is
n
is the purpose of this section to construct a bijection w ~ ~ of the set
= SCml ,
m2 , ••• , m ) onto itself that will have the same properties as
n
the fundamental transformation described in section 1. In particular,
S
\)
holds for all
w in
x,y
(w)
=
S and any couple
Cx,y)
of integers such that
x <y .
x
xm be an element of S and let w = xl x2
m
denote the non-decreasing rearrangement of w. We
(w) is the number of integers i
x,y
x.~ = x , x.~ = y
on the other hand ~x,yCw)
recall that
and
such that
\)
1
~
i < m and
S of the rearrangements of
\)
x,y
(x,y)
and
x •
such that
1
~
i
~
m
is the number of integers
For instance, taking the set
1123 we observe in the following table that
have the same distribution over S for the qrdered pairs
"'x,y
= (1,2) , (1,3) and (2,3). Consequently, the number of exceedances
~
E
=
D
=
I
x<y x,y
\)
and descents
I
x<y
have also the same distribtuion over
~x,y
S •
:0<28-
w
v
t::
12
12
vB
t::
13
v
23
[;23
E
D
1123
0
0
0
0
0
0
0
0
1132
0
0
0
0
1
1
1
1
1213
1
1
0
0
0
0
1
1
1231
1
0
0
1
1
0
2
1
1312
0
0
1
1
0
0
1
1
1321
0
1
1
0
0
1
1
2
2113
1
1
0
0
0
0
1
1
2131
1
1
0
1
1
0
2
2
2311
1
0
1
1
0
0
2
1
3112
0
0
1
1
0
0
1
1
3121
0
1
1
1
0
0
1
2
3211
1
1
I
0
0
1
2
Z
TABLE 3
In the permutation case when we had to define
a
starting with
8,
z, ... ,
w ) of 8
p
and a was simply the functional product of the cycles (wl)(w ) .•• (w ) .
Z
p
For instance, with ~ = 316872495, we obtained the increasing factorizawe considered the increasing factorization
tion
(31, 6, 8724, 95)
and
a
(wI' w
was the product of the disjoint cycles
7248]
(8724
leading to
a
1 2 3 4 5 6 7 8 9)
(.3 7 1 2 9 6 8 4 5
=
In the multipermutation case we try to make an equivalent construction,
although the notion of cycle no longer exists.
~
=
311266387422681959
For instance, let
be a rearrangement of
increasing factorization of
~
is
w= 132332456378292.
(3112, 6, 63, 874226, 81, 95, 9) •
The
Mak-
-29-
king IIcycles i l as above leads to the matrix
'[1123
3112
6,
36
742268
6
63
874226
18
81
I 59
95
Next, if we permute the columns of this two-row matrix in such a way that
the first row is the non-decreasing rearrangement
122332456378292, 'and
the order of the columns having equal top elements is preserved, we obtain
the matrix
1 I 1 2 2 2 3 3 4 5 6 6 6 7 8 8 9 9] .
[ 3 1 8 1 4 2 2 6 7 963 2 861 5 9
- ---As in the permutation case we can associate with
above matrix, say w.
struction of
~ +
w.
Observe that the descents of
~
are mapped onto
It remains to give the construction of
the general case and also that of w +
~,
~ +
w in
which does not seem to be
The present construction was first published in Foata (1965),
then Cartier and Foata (1969) developed a
describe it.
the second row of the
We have just described with an example the con-
the exceedances of w.
straightforward.
~
n~re
convenient algebraic set-up to
Here, a self-contained expos6 is given, based upon the techni-
ques of the latter work.
Let
X be a non-empty totally ordered set.
generated by
the letters
X
We form the free monoid
It consists of all the finite words
xl' x ' ... , xm being elements of
2
'length of the word xl x 2
x
m
X.
xl x 2 ••. x (m > 0)
m
The integer mis the
The free monoid is denoted by
X*.
We
assume that it contains a word with no letter, called the empty word and
denoted bye.
duct and
e
The associative operation in
is the identity element.
that we have to introduce the set
M(X)
x* is the justaposition pro-
It is clear from the above example
of all two-row matrices
(ww'J '
-30-
where
of
wand· w'
x* with the same length.
are two words of
is the length of any of the words
[ww'J
w,w'
simply the free monoid generated by X x X,
more appropriate for our further construction.
position product of two elements of M(X)
[:~] [:~]
Two elements
2
and
[
exist two matrices
with
M(X)
is
but the matrix notation looks
In particular, the juxta-
is given by
=
of M(X)
wW ']
In fact,
The length
are said to be adjaoent if there
2
[~' ] ' ·[vv'J
x, x', y, y'
of M(X)
elements of X
x'
(29)
and two one-column matrices
(~')
such that
y' ,
;t
and
·Notice condition (29): the commutation rule refers only to the first rows of
the matrices; moreover, two adjacent matrices differ br two adjacent columns
whose top elements are di8tin~t. Next, two matrices [:~] and [:~] are
f
l:!]v~zr:~]i~.~~~Y[::j e:::: ::a: 1:!Je=e[:~Js,a[::Ju:nl:~Jf,ma:::cel:~:~]
and
VV
[
!.]
1
are adjacent
for
1
~
i
~
p.
The equivalence relation just
1
defined is compatible with the product in M(X).
M(X)
The quotient monoid of
derived by this equivalence relation is called the jiow monoid and
denoted by F(X)
Its elements are called flows.
The equivalence class
-31-
of an element
[:'J
injection of X x X into
~J
all
with
x,y
in
x an element of X ,
xm
(v ].
X.
[~] + ~]
The map
[~m]
(m > 0)
is an
is generated by the set of
F(X)
v is a word of length m (m > 0)
If
[: 1
~
length of the flow
and
and
has a single representative, namely
is the identity in F(X) .
~ = [:~1 [:~1
Let
F(X) ,
the flow
G
~ee]
The flow
I='l·
will be denoted by
be a flow.
It is called the empty fZow.
The integer m is the
(the length of the empty flow is
Finally, for
0) .
and x,y in X we denote by
nx,y(~)
(30)
the number of i
such that
1
~i ~m
[:~] = ~]
and
•
Before giving the first theorem on the structure of the flow monoid
F(X)
we introduce some additional notation.
belong to M(X)
For any x in
ing sequence-of integers
denote by
i
3 I 1 2 4 1 5)
(:'] = ( Z 3 1 224 1
--
(:'J
8.
o
X
such that
1
=
[xi
z ... x~J
X
z ...
xm
be the increas-
Xl X
(iI' i 2 , ... , i )
p
I ~ i ~ m and x! = x.
1
1
Then
For instance, for
. .. x.
1
we have
-
WI]
[W
X let
x the word x.
o
Let
p
= 314
Of course, the word
is empty if x does not occur in w' .
Proposition.
Let
(1')
(ii)
l
[
WW '
J
and
1
[Ww11'J·
x
[ww2Z' J
2
= [ww 'J
z
The worod WI
be two equivalent elements of M(X) .
o
x
Then
hoZds foro everoy x in X.
(roesp. wi)
is a roearroangement of
W
z
(roesp.
wp .
-32-
PROOF. Properties (i) and (ii) trivially hold when
adjacent.
9.
'
W ]
Wll
[
[:~]
and
are
Hence they are also true for any two equivalent elements of M(X) •
Q.E.D.
Theorem.
~
Any non-empty flow
has a unique faotopization of the for,m
(31)
with xl < x2 < ••• < xn
any peppesentative of ~ ,
>
0, ... , m > O.
m
x n
n
peappangement of w' ,
and
(32)
• X.
fop any
1
i
= 1,
Each flow in the form
single representative
fvx ]
G
with
x in
X and v in x* has a
[vxrnJ . Then, to establish the first part of the
is equi-
valent to exactly one product of the form
a)
(~' )
with v'
Existence of
(~' J .
i. e. to
non-decreasing.
Let
be an element of M(X) .
has length one. Assume now that
w')
is at least two. Denote by wi the longest right
There is nothing to prove when
(W
the length of
(ww')
factor of
having no letter equal to
Wi
is
is the non-deopeasing
theorem, it suffices to prove
a matrix
If
2, ... , m •
rn
Proof.
n
xn
-33-
for some word
and also
and some
y of X.
element
[:')
= [:~) [:0]
But
the
[:~]
matrix
for some words wI' w2
z xn]
[Wi
W
t'tT
w y
l
to
By induction on its length the matrix
(~' ]
to an element
of wI' w2' ,
with u' non-decreasing.
the word
u'xn
has the desired property.
fortiori to
As
is non-decreasing.
is equivalent
2
Wi wz]
[wI Wz
u'
is equivalent
is a rearrangement
The element
Moreover, it is equivalent to
then
u'xn·)
(uy
Wi wz xn]
[wI Wz y
and a
for the equivalence relation is compatible with the pro-
duct.
b)
put
~~!~!~~
m m
Let xl l x 2
2
o x. = v.
[:' ]
1
1
mn
be the non-decreasing rearrangement of w'
x
n
for i = 1, 2, " , n . Further, let
..
(~,)
mZ
x
[ m 2
l
=
::0] .
:~ Vz
Then we show that any element
(~' ]
non-decreasing, is necessarily equal to
ro
m2
l
8 (ii) that t ' is equal to xl X
z
rearrangement of
W'
and
•
Hence
(~' ]
equivalent to
v'] .
(V
x:n
[:,]
, with t'
It follows from proposition
which is the non-decreasing
is equal to the product
some words
is equivalent to
But
(~']
0
xi
=t i
proposition 8 (i) implies
-34-
that
[:,)
Hence
t
xi =
(~,)
= t l t z ...
0
tn
i.e. v.1 = t.1
xi
= vI
v2
v
n
=v
for any i
= I,
2, ••. , n
Q.E.D.
•
10. Corollary.
The cancetzation lCOJ) holds in the flow monoid
foT' any fiows
implies
Proof.
4>
= </>'
Let
the equality
</>, </>', </>",
</>
</></>"
F (X).
= </>'4>"
That is to say"
(T'esp.
4>"4>
= </>"</>')
•
x,y two elements of X,
be a flow,
and consider the
equation in 1/J
</>
Using the factorization of </>
= 1/J~] .
given in theorem 9 we have
But this equation has a solution only if x is equal to some Xi (1 s i s n) •
If x
+=
= x.
(say), then v.
l
[:~r .. [::C
r ... rl:
l:~
l
m]
] '"
m. -1]
1
•••
I::n]
rJ
m
l::n
= wy
for some word w.
~J .
It follows from theorem 9 that
defines a unique element
then conclude that the equation (in 1/J)
has at most one solution in F(X)
But we also have
W of
F(X) •
We
-35-
Consider now three flows
<jl
= cp'
<jl"
cp"
and
cpi
et>,
As
<jl
~l = <jli
GJ
implies
for any x,y
in X,
we conclude by induction on the length of
= </>'</>11
implies
</>
that
<jl<jl"
= cp'.
The "resp." part is proved in
Q.E.D.
the same manner.
Note that in any element
of H(X)
(ww')
the word w'
is not neces-
sarily a rearrangement of w (although it is of the same length).
it is a rearrangement, the equivalence class
a airauit.
Clearly, the set
C(X)
of
(ww')
When
is called
of all circuits form a submoniod of F(X) •
If follows that from theorem 9 that any circuit y has one and only
where -v is non-deoreasing and is a
one representative in the form
rearrangement of v.
We then define
(33)
(34)
row matrix of the form
Designate by
=v
f(v)
(~
~
fVvJ·
~
We then have
and v in X*.
for any y in C(X)
IT: C(X)
there corresponds one and only one two-
with v the non-decreasing rearrangement of v •
the airaui t
(35)
v
=v.
.IT(y)
Conversely, to any word v of x*
nf(v)
=
y
X*
and
f:
are bijeative and inverses of each other.
(36)
TI(y)
=y
and
few)
X*
~
fn(y)
=y
and
Thus the two maps
C(X)
Moreover
= w,
denoting again by -w the non-decreasing rearrangement of a word w.
definition of
f
is straightforward. 'On the other
obtained as follows:
take any representative
(ww,)
hari~
the word TI(y)
of y and let
The
is
-36-
Xl' x2' .•. , xn be the only letters occurring in w' (or w) • Then
IT(y) is the word vI v2 ••• vn where vi (1 s i s n) is given by formula (32).
The final step is to define another bijection
~:
x*
C(X),
+
and
the fundamental transformation w + Q will be the functional product
For any word w = xl x2 ••. xm we denote by
~w
the word
As in section 1, we say that a non-empty word is
initia~~y
~
-1
0
(37)
dominated or sim-
ply dominated if its first letter is (strictly) greater than all its other
letters.
In the same manner, a circuit y is dOminated
w dominated.
Clearly, for any circuit
word w such that
c(w)
y
= [~w].
i~
y
= [:w]
with
y there is at most one dominated
When w is dominated, we will denote by
the circuit
(38)
Thus
c
is a bijection of the set of all dominated words onto the set of
dominated ciruits.
Let
y
= [~wJ
be a dominated circuit. Then the first
letter of w (that lidominates" all the other letters), formerly denoted by
Fw,
11.
will also be written Fy.
Proposition.
Any non-empty circuit admits exactly one increasing dominated circuit
factorization.
one sequence
That is to
say~
(YI' Y2' ... , Yr)
to any circuit
y
there corresponds exactl.y
of dominated circuits suah that
r .
-37-
Before proving proposition 11 we establish the following lemma.
12.
Lemma.
If for i > 0
of X and a matrix
(i)
(ti)
Zj
~
m·
th[v~:1.!'e]
zo for
exists a sequenae
of M(X)
suah that
[:~] [:~_1 ::: :~ :~]
then an element z. 1 of X and a matrix
1.+
(E)
of elements
0 < j s i
i. equivalent to
so that aondition
(zo' zl' ••• , zi)
holds when i
[Vi+l]
vi + 1
of M(X)
is replaaed by i
aan be found
+ 1 •
as the bottom element of the rightmost one-column
letter equal to
element
Thus, condition (ii) also holds when
We then
Z.
1.
is then equivalent to
i
is replaced by
i + 1 •
We are now ready to complete the proof of proposition 11.
Q.E.D.
We first
prove the e:x:istenae of an increasing dominated circuit factorization by
-38-
showing that any non-empty element
(~
of H(X)
I
with -v
the non-
decreasing rearrangement of v I is equivalent to a product of the form
OW
owr
OW
[W
[w ] with wI1 wZ1 •.• , wr dominated words and
[W
z
r
FW l ? FW2 s
Such a product will be called an inaX'easing dominated
II] 2]
0
0
0
pX'oduat.
is of length one, then
Let
is equivalent to (~. =
m mZ
m
x n (xl < X <
v = y' y' .•• y~ = xl l X '
< x ,
I 2
z
n
z
n
m > 0, m > 0, •.. , m > 0) and v = YI Y2 . •• y , with m > I
2
l
n
m
Condition (ii) of lemma 12 holds for i = I if we take v' = y' Y'
Y~-l
1
I Z
vI = YI YZ
Ym- l , Z0 = y'm (= xn ) and zl = Ym • If Ym = y'm (= xn ) ,
If v
...
(~
[~v) .
...
.
...
...
then
(~
is equivalent to
By induction on the word length
is equivalent to an increasing dominated product
As
xn is greater than or equal to any letter of vI' it is also greater
than or equal to any letter of wI W ... w • Hence the product
r
z
[WOWI I]
n
[wowrr] [xx ]
n
is an increasing dominated product, equivalent to
z I = Ym ;t; y'm = z0' then conditions (i) and (ii) of lemma 12 both hold
for i = 1. By applying lemma 12 inductively we can then form a sequence
If
(ZO' zll •.. )
of elements of X. Let i+1 be the first integer for which
lemma 12 does not apply.
Such an integer exists since the sequence
·is necessarily finite.
still
for
0
< j
s i.
We then have
If i + 1
<
z.1+ I
m the word
= zo'
(~
but
is then
,
-39-
equivalent to
and the word w = Zo zi .•. z2 zl
is dominated.
Again by induction on its
length the non-empty matrix [Vi+l] is equivalent to
product
[~:l] ... [~:r]
is equivalent to
m
the inequalities FW
l
(~
~
is f3.quivalent to
an increasing dominated
co::::uently, the product
Moreover, as
...
~
Fw = xn (the maximum letter of v) ,
Fwr s Fw hold. Finally, it i + I = m , then
[zm-l zm-Z ...
zl zo]
...
Zz zi
Zo
zm_l
, and the word Zo zm_I'"
z2 zi
is itself dominated.
Assume that the two increasing dominated circuit products
13
1
6
2
f3
q
and YI YZ ... Yr are equal. Furthermore, let v!' v2 ' ... , vq ' WI' w2 ' .. .,w
be the dominated words defined by (31 = c(v l ) , 82 = c (v z), ... , aq = c (vq ) ,
c(w)
(see (38». Also put
r
vq = YI Y2 •.. Ys
= [;~ :::
~:_:lYl]
and wr
= zl
and Yr·
Zz ... i t '
[~:rl · I:~
t
6q = fVovqq]
so that
: : ::_:lZtl' But Yl (resp.
is equal to the maximum letter of VI v 2 ••• vq (resp. WI w2
wr ). As
the two words VI v 2 ..• vq and WI w2 ..• wr are rearrangements of each
other, we have YI
= zl
Hence Ys
Ys - I = Zt_1 ' ..• , Yl = Zt-s+l'
But
= Zt'
Assume
YI (= zl)
5
st.
By induction
is the maximum letter of
-40-
both vq
and wr
As wr
Zl
= Zt-s+l
can
q = wr and Sq = Yr • As the cancellation
law holds in F(X) (see corollary 10), we obtain 8
8q _l = Yl .•• Yr-l
1
and the unicity follows by induction on the length.
Q.E.D.
hold only if t
=s
is dominated, the equation
Hence
V
(wI' wz' ... , wr ) be the increasing factorization of a word
w = YI yZ .•. Ym' As we have seen in section 1, the factors wI' wz' ... , wr
are initially dominated words and FW I S FW Z S
We then form the
Let
,
dominated circuits
C(X))
... ,
Taking their product (in
we obtain the circuit
(39)
Remember the definition of
~x,y
(given in section 1) and
nx,y in (30).
13. Proposition.
The map 6 is a bijeation of x* onto C(X) 1JJith the foZ"lo1JJing ppopepties:
(i)
(ii)
6(w) =
w fop
aZ"l w in X*·,
fop any x,y in X suah that x
(40)
Proof. As we already know c:w
< y
1JJe have
= nx,y (6(w)).
-+
[:w]
is a bijection of the set of
dominated words onto the set of dominated circuits.
Let
(WI' wz'
... ,
wr )
-41-
be the increasing factorization of a word w.
As
FW
1
FW
~ ..• ~ FW
'
2
r
transforms the
~
(wI' w2 ' ... , w ) + (c(w l ), c(w 2 ), .•. , c(wr ))
r
increasing factorization of a word into the increasing dominated ciruit
the map
factorization of the corresponding circuit.
sition 11 that
6 is bijective.
It then follows from propo-
Property (i) is trivially verified.
It
remains to prove (40).
w = YI Y2 ... Ym be a word. Denote by (WI' w2' ..• , wr ) its
increasing factorization and by iI' j2' ••• , jr the indices of the outLet
standing letters in w (see lemma 1).
-- y'1 y'2 ' " y'm'
By convention, let
··[Yl Yi ...
k
is dominated.
oW
2
•.• oWr
=
J-r+l =m+l.
= 1,
2, .•. , r
Y~J
Yl Y2 ... Ym
When the index
J·2-1.J·
. 3 ·-1.. ••. , J' r+1 - 1
hand, for any
= OWl
Then
6(w) =
from
Finally, let w'
we have
Y!
i
[m]
of
= Yi +l
.
is different
On the other
because
we have
Finally
Y
jk+l- l
~
y.
Jk+l
for
k
= 1,
2, ..• , r - 1
because y.
is an outstanding letter. Take then two elements x,y of
Jk+l
X such that x < Y . From the previous remarks we have y!1 = x , y.1 = y
only if Yi = x Yi +l = Y
i = j2 - 1, i - 1, .•. , jr+l - 1.
3
if
an~
Furthermore this cannot hold when
Thus there are as many couples
(y!, y.)
1
1
-42-
equal to
(x,y)
as there are occurences of the factor yx in
w = Y1 YZ ... Ym '
Q.E.D.
This establishes (40).
Denote by ~-1: C(X)
+
the inverse of ~.
X*
If w is a word we
put
(41)
~
-1
(f(w)).
14. Theorem.
The mapping w +
Zowing
~
is a bijection of X* onto itseZf having the foZ-
p~ope~ties:
(i)
(ii)
~
is a
vx,y (w)
~e~~angement
= ~ x,y (~)
of w
fo~
any x,y in X such that x
< y ••
Proof. As both f and ~-1 are bijective, w + ~ is also a bijection.
Property (i) follows from (36) and proposition 13 (i).
w = Y1 Y2 ... Ym be a word and w = Yi Y
rearrangement.
relation
Then
vx,y (w)
few)
z '" ..
is the circuit
= nx,y (f(w))
for
x
<
r
y'm be its non-decreasing
Y YZ
11'] ~ ]
y
~
Now let
fy2
rl
Ym
V
A
V
A
V
'
y
']
The
m
Y is then a trivial consequence
of the definitions of vx,y and nx,y
From (40) we deduce that
t;
x,y (~) = nx,y (~(~)) = n.x,y (f(w)) = vx,y (\q).
Q.E.D.
-43-
5.
Recapitulation of the previous section and intercalation monoids.
Let us sketch again the construction given in the previous section.
We
have considered:
X a totally ordered non-empty set;
X*
the free monoid generated by X;
M(X) the set of
~o-r~
matrices
(ww')
with w,w'
two words of x*
of the same length;
F(X)
the flow, monoid, i.e. the quotient monoid'of M(X)
derived. by the
adjacency relation;
form
(:,)
[:']
the flow associated to
C(X)
the circuit monoid" the submonoid of F(X)
[:']
with
Wi
;
made of flows in the
a rearrangement of W;
The bijection w + ~ of x* onto itself is then the product
where A and r- l are defined as follows:
r: X*
+
C(X)
is the biJ'ection defined by
few)
= Tw]
Lw '
A-lor
where -w is
non-decreasing rearrangement of w;
0: X* +
x* is the cyclic left shift defined for any word
w = Yl Y2 .•. Ym by ow = Y2 Y3 ... Ym Yl ;
A: X* + C(X) is the bijection defined for any word w,
inareasing faatorization
A-1
(w l , w2 ' .,., wr ) ,
is the inverse of A.
by
with the'
-44-
To obtain 6- 1 (y)
starting with a given. circuit
y "we have to form
(Yl' Y2, ..•• , Yr ) of
is the .juxtaposi.tion product wI wi ••• wr ' where \~i
the non-deareasing dominated airauit faatorization
Y.
Then 6
-1
(y)
is.the unique dominated words defined by
~
ow.]
wi
1
= y.1
for each i
Let us take again the same example as in section 4.
few)
= 1,
i, ... , r
The word
is the circuit
f1 1 1 2 2 2 3 3 4 5 6 6 6 7 8 8 9 9]
l1. 1 ~ 1 i
2 2
~ '!... 2.
6 3 2
!
6' 1 5 9
•
The non-decreasing dominated circuit factorization of proposition 11 may be
obtained for
few)
by "sorting out" the successive dominated circuits from
right to left:
few)
eJ
= [;
1 1 2 2 2 3 3 4 5 6 6 6 7 8 8 9]
1 8 1 4 2 2 6 7 963 2 861 5
= [;
1 1 2 2 2 3 3 4 6 6 6 7 8 8] '[59]
95
1 8 1 4 2 2 6 7 6 3 2 8 6 1
=
=
g
1 2 2 2 3 3 4 666
1 1 4 2 267 632
g
61
1 2 3 3 6
1 1 2 6 6 3
e
.
~ ~J
9
'eJ
9
nn g~J [~l
6 L8118]
4 2 2
8]
8 7 4 2 2 6.
[59] [9}
95
9
= [1
1 2 3 6] f36]. T7 4 2 2 6 8] fIB] fS9] f9]
3 1 1 2 6 L63 L8 7 4 2 2 6 LSI L95 L9
1 2 3] r6] r36] r7 4 2 2 6 8] fI 8] f59]
3 1 1 2 L6 L63 L8 7 4 2 2 6 L81 L9S
= [1
Th~s
T91
L9
product is the non-decreasing dominated circuit factorization of few) •
The word w = 6- 1 (r(w)) is then the juxtaposition product of the words
-45-
occurring in the second row of the last product, namely
~
=3
The exceedances in
Now Let
words
w,
Wi
= (X*
r(w)
and the descents in
n: C(X)
+
of x*
the formula
product in X*,
CI(X)
1 1 2 6 6 3 8 7 4 2 268 1 959 .
, T)
have been underlined.
~
X* be the invepse bijection of r .
W T Wi
= n(r(w)
r(w l ) )
called the intepaaZation ppoduat.
defines a new
The ordered pair
is said to be the intepoaZation monoid.
n is a monoid isomorphism of C(X)
For any two
By construction
onto C'(X) .
15. Proposition.
of x* denote by xl' x2 ' •.. , xn the inopeasing sequenae of the Zetteps oaaUPPing in eithep w OP Wi • Let mi (pesp.
Given two 1JJopds w, wI
mil be the multiplioity of xi in w (pesp.
2' ... , w~))
let
(wI' w2 ' ••• , wn ) (pesp. (wi, w
w 1JJith w.1 (pesp. w!)
of length m1·
1
W T Wi
= W1
Wi W
Wi
122
W Wi
n
n
(pesp.
(1 ~ i ~
Wi)
FinaZly~
be the faatopization of
m!)
(1
1
~
i
~
n).
•
Proof. The non-decreasing rearrangement of w (resp.
ml .
n
... xn ) .
Thus
n) .
Then
r (w)
=
rlW;!
Wi)
is
x
I
and
Then
-46-
l
m +mn]
Xnn
..."
Consequently
W 1 WI
= IT(r(w)
r(w l ) )
W WI
n
= WI
n
wi ••.
Q.E.D.
W w~
n
16. Example.
Then
= 31 • e . 1
= 31521245443
W
W 1 WI
• 45 • 4 and
WI
=e
. 52 • 2 . 4 . 3.
Hence
.
We now make use of the intercalation product to give a non-commutative
generalization of the "Master Theorem" of MacMahon ((1915), p. 97).
The.
Let Xl' ••• , Xn be n aommutative
indeterminates~
B = (b ) a squaroe matrix of order n bJith real- entl'ies~
ij
and Y. = 2 b.. X. . For any sequenae of n non-negative integers '
1 l~j~n 1J J
m
m
1
••• , mn the aoeffiaient of the monomiaZ Xl •.• xnn in the poZynomiaZ
''Master Theorem" is stated as follows:
m
• •• Y
n
n
is equaZ to the aoeffiaient of the same monomiaZ in the
e~pan-
sion of the inverse of the determinant
O. 0
(42)
D=
o. o'
000
- bIn Xn
00 0
oool_b
X
nn n
in the formaZ pObJer series aZgebl'a bJith n indeterminates Xl' ••. , Xn
-47-
Now for any i
= 1,
2, •.• , nand
m ~ 0 we have
... + b.1n Xn)m = l ' L . b..1J ... b..1J X.J
m
l
l
s;Jl,···,Jms;n
+ ... + m
.tr the sequence
n and denote by .tl !2
m
m
m
n n . Hence Y l ... Y n =
... b.t k
b
L
l
n
.tIkI
~= (b i l Xl +
1
Let
ml
1
r = ml
m2
2
X.
Jm
X
kI
r r
lS;kl, .•. ,k s;n
m
ml
m r
X n in Y
y n is then equal to
n
n
m
The coefficient of Xl l
(43)
where the summation is extended over all rearrangements
m
n n .
sequence
We let
k l ••. kr
of the
T be the formal power series
m
n .
X n
(44)
Thus the "Master Theorem" asserts that the two formal power series
T and
D (reduced to a polynomial, see (42)) are inverses of each other, i.e.
(45)
T D
=
l.
Before proposing a non-commutative generalization of this identity let
us further transform the expression for
matrix
diag(X l , .•. , Xn )
D.
Let
~
denote the diagonal
and 1 the identity matrix of order n.
Then
D = det (I - B!) .
Now, if U is a square matrix of order n,
the development of det (1 - U)
in terms of the minors of U along the diagonal is well-known.
This develop-
ment applied to U = B! yields
det (I - B!)
= o<_rL<_n
L .
(_l)r.
1 <···<l
l
r
det (bkl X.e? (k, .edi , •.• , i }) .
l
r
Xk
I'
Let S(il, •.. ,i )
r
denote the permutation group of set {iI' ••• , i r }.
Then
D = det (I - B!)
(46)
=
where the third summation is extended over all
0
consider n non-commutative variables
how the product of two monomials in the x.1 's
is to be taken).
write a formal series having the same expression as
which
Xi
(47) d =
is replaced by Xi (i
r
; <
0 ~r_n
r
l:
(_l)r • .
1 < ••• <l
1
= l, ..• ,n)
r
In the same manner, the formal power series
D
We could then
(given in (46)), in
. We obtain
e:(o) b.10 (.1 ) ••• b.1
1
in S(il, •.• ,i ). Now
r
(we shall soon explain
C·)x (. ) ••• x0 C'1 ) •
r
rO 1r 0 11
T given in (43) and (44) can
also be written
i l •••
Again we form
(where
~
denotes the non-decreasing rearrangement of kl ••• kr ) •
(48)
Now the following question arises:
bra in the variables
(49)
does there exist a non-commutative alge-
Xl' •.. , xn '
dt
in which the identity
= td = 1
holds, with the additional property that a homomorphic image of (49) is sim-
-49-
= TD = l?
ply the "Master Theorem" identity DT
With the help of the
intercalation moniod we can build such an algebra as we now show.
Let
order
xl
X be a finite set
xn '
form the set of all formal- senes
<
x2
X = {xl' x2 ' •.. , xn} (n > 0) with the natural
Given a commutative ring with identity n we
< ••• <
a =
L a(w)
w,
For any word w the coeffi-
where w runs over all the elements of X*.
cient
a(w)
is an element of n.
The intercalation product for words is
now extended to the set of formal series.
(L
(50)
a(w) w) T
(L
b(w) w)
We let
= r c(w)
w,
where for any word w
c(w)
(51)
= r a(w
t )
b(w") ,
the summation being extended over all words
course, there are finitely many words
the above definition (50) makes sense.
Wi, w" with WiT wI! = w.
Wi, w"
defined in (47) and (48) belong to
Wi T w"
With the multiplication
of formal series is a ring we will denote by
t
such that
= wand
T the set
n(n,T) • The two series
n(n,T).
Of
d and
It is our purpose to esta-
blish the identity
d
(52)
For any formal series
a
T t
in
=
t T
n(n,T)
d
let
=
I.
~(a)
denote the series obtained
... ,
Xn for xn
Denote" by n(n) " the usual
formal power series algebra in the n commutative indeterminates Xl'
, Xn
with coefficients in n . Clearly iP is a ring
by substituting
Xl
for
xl'
...
-50-
homomorphism of O(n,T)
into
O(n) • Furthermore
wed)
=D ,
and so applying w to (52) yields MacMahon's identity DT
=T
~(t)
= 1.
We then
have a non-commutative generalization of the "Master Theorem".
Since
is commutative, the formula
0
(53)
C(X)
defines a homomorphism of the circuit monoid
monoid of O.
monoid C(X)
Hence
g
= for
is a homomorphism of the intercalation
into this multiplicative monoid.
g(w)
w = xl
Xl
be the
r
Then
= f( [:])
Accordingly, 'the series
(55)
Let
1
non-decreasing rearrangement of w =
(54)
into the multiplicative
t
can be rewritten as
t =
r g(w) w •
The coefficient of the word w = Xa (i ) ••• Xa(i ) in the series d (given
r
1
r
r
in (47)) is (-1) e:(a) b.1 a CO)'"
b.
C')
=
(-1) e:(a) g(w). Let p' (a)
1
1 a 1
1
r
1
r
denote the number of orbits (or disjoint cycles) of the permutation a. From
the definition of the signature
(_l)p'(a)
e:(a)
it follows that
= (_l)r
e:(a) •
We can then put
(56)
~'(w)
= (_l)p'(a)
.
-51-
For the empty word
l.I t (e) = l .
e let
Finally, when w hasrepeated
letters, let
(57)
Notice that the only words of x*
{xl, •.. ,Xn })
(given in (47)).
d
(58)
Now the product
d Tt
(d
= L l.I t (w)
As
g(w)
= L a(w)
with a(w) = L<d(w t ) t(w"): wt or w"
= w}.
Hence d can be rewritten as
W •
given in (58) and
d T t
T W"
X is the finite set
(remember that
with no repeated letters are the words actually occurring in
the formal series d
wt
=0 •
1.I t (w)
= w}.
t
in (55)) is equal to
w,
Hence a(w)
= r {l.I t (w')
g(wt).g(wl.')t
g is a homomorphism, we obtain
a(w) = g(w) L{l.I'(W t ): wt T wI!
In the same manner,
t Td =
L at (w)
= w}
with at (w) = g(w) L{l.I t (w~'): wt T w" = w}
Thus the two identities (52) will be established if we show that the following
identities hold:
wt TW"
= w} = L{l.I'CW
I
IV
):
WtTW"
= w} = { 0
if w = e
otherwise. Let
C'1. ) be a word without repeated letters. As above, denote
r
x.
its non-decreasing (in this case increasing) rearrange-
w = xo (.1. ) •.. x 0
1
by
w= x.
1
1.
1.
r
[x..
..
x.
= few) =
r
.1
1
1
1
is called eZementary.
xoCil ) •.• Xo(i )
r
By convention the empty circuit is also elementary. If the .
mente TIlen the circuit
y
-52-
permutation
r
a is the product of the p cycles
z ••.
k
with k.
J1
k j 1-1 k 1
]
1 ••. k.J1- 2 k.J - 1
1
< ••• < k.
Jp
00 0
r
Jp
k.
Jp
k
r
+1
k
r-1
k.J p]
,
k
r
then the increasing dominated circuit factorization
of Y is
X k1
]
.., 0 0
x
x
k.J - 1
1
Let p(y)
k.J
• •.
p
~r
x
kr- 1
denote the number of factors of the increasing dominated circuit
factorization of y.
Then
(60)
When p(y)
[xkJ.p+1
p(y)
= 1,
the circuit
= p'(a)
.
y is called a aycZe.
It then follows from
proposition 11 (but also from the elementary theory of the permutation group!)
that any elementary circuit has a unique increasing cycle factorization.
particular p(y)
is the number of cycles in this factorization, in short the
number of its ayaZes.
(61)
Let
p = p'
.0
r .
From (56), (57), (60) and (61) we deduce that
(62)
if the circuit
(63)
In
p(y) = (-l)P(y)
y is elementary (in particular
p (y) = 0
p[:}
= 1),
and
-53-
otherwise.
(64)
To establish (59) it then suffices to show that
I if y is empty
= {0 otherwise
2hl(y'): y'y"= y } = l<1J(yl'): y'y" = y}
If y = [:]
the above two summations are equal to 1 by definition of 1J •
It then remains to'establish (64) in the non-empty circuit case.
y'y" = y we say that
y.
y'
We denote by L(y)
(resp.
(resp.
elementary factors of y.
circuit is elementary.
y")
is a Zeft (resp. !light)
R(y))
When
factor of
the set of all left (resp. right)
Neither of these sets is empty, as the empty
By taking (63) into account we are reduced to show-
ing that for any non-empty circuit y we have
l<1J(y'): y'e:L(y)} = 2{1J(y"): y"€R(y)} = 0 •
(65)
Let us denote by L'(y)
consisting of oyoZes.
(resp.
R'(y))
the subset of L(y)
(resp.
R(y)
Then identity (65) will be a consequence of the fo1-
lowing lemma.
16.
Lemma.
For any non-empty oircuit y
empty.
Furthermore
L(y)
(resp.
distinct cycZes that beZong to
the set
R(y)
L'(y)
Assume that lemma 16 is proved.
distinct
r
cycles
L(y)
= {Yi
1.
Yl' Y2'
"'j
(resp.
(resp.
is non-
R'(y)) .
Suppose that
j
R' (y))
is the set of aU products of
Yr (r > 0) .
•.• y. : 0 S P S r
J. p
L' (y)
L'(y)
consists of the
Then
1 S i l < ••• < i p S r} •
-54-
It follows from (62) that
II (y.
1
for any sequence
[~]
1 <- 1. 1 < ... < i ~
P
having r cycles we deduce that
such that
L(y)
•.• y. ) = (-l)P
1
p
1
As there are
r
l{ll(y'):
Y'~L(y)} =
l
Osp~r
l hi (yll):
A similar proof holds for
(_l)p[r]
p
ylie:R(y)}
P
elementary circuits in :,
= (l-l)P = 0
=0
i
i1
•
•
Lemma 16 can be deduced from a theorem on the structure of circuit
monoids (see Cartier and Foata (1965), Chap. 3).
The function
II
is the
celebrated Mtlbius function of the circuit monoid and the identity of lemma
16 can so be derived from general properties of Mtlbius functions.
However,
in one way or another the formula for the Mtlbius function has to be found.
The proof of lemma 16 given here involves, at least implicitly, the calcu-
lation of the Htlbius function of the circuit monoid.
As any circuit is the product of dominated circuits (proposition 11),
the sets
L'(y)
and
R'(y)
will be shown to be non-empty, if we prove that
any dominated circuit can be expressed as a product of cycles.
proved in the following way.
with Fw
in
=
z.
Consider a dominated circuit
This can be
[~w],
with
If it is not elementary. we have w = zW l xw z xW with
3
x
x,
has no repeated letter.
But
since the words
and wzx have no letter in
If
¢olll.mon.
we have
= w'yw"
Z 2
for some words
y
is the maximum letter of
As
fW2XJ = fW2WZyJ '
Lxwz
Lrwzw
z
-55-
the circuit
[:;:]
is dominated and elementary, and so is a cycle.
is the product of a dominated circuit and a cycle.
Hence
By induction on
the circuit length, this shows that any dominated circuit is a product of
cycles.
Hence the sets
L'(y)
mId
R'(y)
are non-empty.
Next, for an elementary circuit to be the product of cycles of
it is necessary that any two distinct cycles of
connnon.
L'(y)
TIlis can be proved in the following manner.
with yi, y' in
2
L' (y)
.
If xl
. [:~ ...
xm
x
m-l
have no letter in
Let y
= y'I
ytl
I
= y'y"
2 2
is a common letter of both Yl' yi '
can write
y'I
L'(y)
:~]
and y' 2 -
12 ...
Yn
we
Y1]
YI"· Yn-l Yn
YII y"I . = y'2
y" we 2
necessarily
have x = y n • But
m
again xm-l = y n-l' and so on. As the words
have no repeated letters, we conclude that yi =
Hence any two distinct
As
yi.
cycles of
L'(y)
are actually disjoint.
Finally, we show that
elements of
L'(y).
of L'(y).
L(y).
p
=I.
LI(y).
y'
yi ••• yp
belong to
Conversely, let
We proceed by induction on p.
Assume
p > I
Then y
Also
y
L(y) • Then
y'
is a pro-
and clearly each cycle in this proyi, .•. ,yP be distinct cycles
It remains to show that the product
y" by induction on p
Let
is the set of all products of distinct
First, let
duct of disjoint cycles
duct belongs to
L(Y)
= y'1
= y'P
yi ... yp belongs to
There is nothing to prove when
y'
yli holds for some circuit
p-l
S" since yp belongs to L'(y)
Yl .•• yq be a factorization of yli
into cycles.
We then have
-56-
YI' ••. Y'p-l YI •.• Yq
= Y'P
all
IJ
=Y •
Let y.
J
be the cycle with smallest
We can then write
subscript having a letter in common with Y'
P
Xl]
X2 ••• x
X . yJ.••• yq
[ Xl'" xm
m_l m
with Xl
= YI
• As the product
= [ Y2
Y
.I
Yi ••• yp-l YI"'Yj-1
Yn
YI] B"
Yn- I Yn
does not contain any let-
we have xm = Yn ' Again, as the words xI."xm and
YI"'Yn have no repeated letters, we conclude that xI,,·xm = YI"'Yn and
so y. = Y' • By construction the cycle Yj has no letter in common with
ter equal to Xl'
J
P
••• , y. I .
,
J-
Hence
y1···yp - 1 YI"'Y j - 1 Yj Yj+I···Y q = Yi···yp-l Yp Yl"'Yj-l Yj +1 ···Yq •
p
Accordingly, the product Yi'.' Y
belongs to
L(y) •
The "resp." part of the lemma is proved in a similar manner.
In Cartier-Foata (1969) (Chap. 5) one can find another non-commutative
generalization of the "Master Theorem" identity.
The formula holds in a for-
mal series algebra with coefficients in a non-commutative ring, but with commutative indeterminates.
-57-
6.
The Fundamental Transformation of the Symmetric Semi group.
In this section we extend the definition of the fundamental trans-
formation to the set
Fn of all maps of
is sometimes called the symmetric semigroup.
nn elements
f
of F
an ordered pair
f(x).
The set Fn
As is well known any of the
may be represented by a linear graph with n
n
labeled points as follows:
x,
into itself.
[n]
a fixed point
(x, f(x))
with
x
f(x)
~
f(x)
=x
is a sling at point
is a line from
x to
This graphical representation will be of great help in the descripn and x to
We let f 0 (x) = x and fk(x) = f(fk-l(x)) for any
j-l
f is a sequence (x , f(x) , ... , f
(x» of
tion of the fundamental transformation.
the interval
k > 0 •
[nl
.
A cycle of
j ~ I
distinct elements with
and
is said to be recurrent (for f)
If x is recurrent,
f(x)
Let
f
=x .
fk(x) = x
fj(x)
if
belong to
F
The element
x of
holds for some k
is also recurrent, since
fkf(x)
[n]
~
I .
= ffk ex ) = f(x)
The set of all recurrent elements will be denoted by Rf • Suppose n > I
and take x in [nl • The sequence of the (n + 1) elements
(x , f(x) ,
... ,
~(x))
(0 s i < i + j s n)
Thus
f n-l(x)
contains two equal terms, say fi(x)
As
n" I - i
1.·s always recurrent.
~
0
and
j > 0
and fi+j(x)
we have
Converse ly,· 1f fk()
X
=X
hold s fo r
some k ~ 1, we write n - 1 = qk + r with 0 s r < k. Then
f n - l fk-r(x) = fqk+k(x) = x. This means there is an element y,
fk-r(x)
with the property that
f
fn-ICy)
= x.
n-l ([nD = R .
f
Consequently
namely
.
-58-
In particular
i.e.
=x
fk(x)
fk fk-l(x)
Rf
is never empty.
for some k ~ I,
= fk-l
and verifies
On the other hand, if x is recurrent,
fk(x)
f(y)
let
y
= fk-l(x) = y.
= fk(x) = x.
= fk-l(x)
Then
fk(y)
=
The element y is also recurrent
This shows that the restriction of f
to
f is surjective. As Rf is finite, this restriction, which we will herej-l (x))
after denote by wf ' is a pe~tation of Rf . A cycle (x,f(x), .•• ,f
is isolated if any relation
R
= fk(x)
fey)
(43)
implies
y
= fk-l(x)
.
with
k ~ I
In other words, the cycle
(x, f(x), ... , f j-l (x))
is isolated if (43) implies that y is an element of the cycle.
the cycle
(x, f(x), ..• , fj-l(x))
Conversely,
is not isolated if (43) holds for some
element y that does not belong to the cycle.
Such a y is then non-
recurrent.
Otherwise,
Hence y = f i-I fey)
fi(y) = y would hold for some i ~ I .
= f k+i-l (x)
and y would belong to the cycle.
The set
is denoted by Zf'
[n] \ fern])
always empty.
If Zf
= ~,
The intersection Rf n Zf is
is a permutation of [n]. In the
then f
next lemma we show that any element of a non-isolated cycle can be reached
from some element of Zf by iterating
15.
Lemma.
Let x belong to
[n]
f
sufficiently many times.
and f
be a map of [n]
If x does not belong to an isolated oycZe of f ,
fop some m ~ 0 and z in
Proof.
Xl
~
fexl)
= x.
As
we have fil(z)
=x
Zf'
First assume x non-recurrent.
x with
into itself.
If it is not in Zf'
x is non-recurrent,
Xl
there exists
is also non-recurrent.
-59-
(x o' xl' x2 ' •.• ) of non-recurrent
= x. 1 for i <:= I .
elements can be constructed so that Xo = X and f(x.)
J.
J.By using this procedure a sequence
As these elements are all distinct, the sequence is necessarily finite.
Thus, there exists m <:= 0 such that
is empty.
Zf and fm(xm) = Xo .
Next, suppose x recurrent.
(x, f(x), ••• , f j-l (x))
to
containing x.
be the cycle
As the cycle is assumed.; to be non-isolated, there exists
a non-recurrent element
x
Let
Hence xm belongs
= fj-k+l(X O) '
Xo with f(x o)
= fk(x)
and
1 S k S j.
As
Q.E.D.
the first part of the proof applies.
The previous lemma shows that the linear graph associated to a map f
has one or more connected components, each of which contains a single cycle
of length (number of lines in the cycle) at most n,
a sling counting as
a cycle of length 1.
16.
Example. Consider the map
x
=1
f(x)
=4
of {I, 2, .•• , 20}
2 3 4 5 6
7 8
9 10 11 12 13 14 15 16 17 18 19 20
11 9 9 8 4 15 8 13 16
into itself.
8 19
6 1 9 20
2 7 1 10
Its associated graph is then
I
17
/'~\10
5
/
16--
.
\
\
·~20
-60-
We have Rf = {4, 6, 8, 9, 10, 13, 16, 20}
Rf is the permutation
4
[
6
8
9 10
13
948
and the restriction of f
13
16
6
The fundamental transformation of F
16
20]. •
10
20
will be a bijection f
n
to
+
f
of Fn onto itself, whose restriction to the permutation group Sn is the
fundamental transformation of Sn described in section 1. Accordingly f
has been constructed when f belongs to S
Assume f to be in Fn \ Sn •
n
Then the set
Zf is non-empty.
sequence of its elements.
••• J
W
P
(p
= card
We let
(zl' z2' •• , zp)
We define a sequence of
Zf)
(p
+
be the increasing
1)
words
by induction in the following manner.
First
(44)
that is to say,
o is the image of the permutation
W
mental transformation of the symmetric group.
w under the fundaf
In particUlar W is a
o
rearrangement of the recurrent elements of f. Assume that wo' WI' ••• , wk _l
have been defined for some k with
1
S
k
S
p.
We then let
(45)
where mk is the smallest integer m > 0 such that fmCz ) is equal to
k
w _ • This definition makes sense, because
a letter of the word Wo WI
k l
(42) implies that
is in Rf
and so equal to a letter of W
o
It will be shown in proposition 19 below that the juxtaposition product
o wI
W
(46)
fn-l(Zk)
••• wp has length n.
Accordingly we define the map
f by
-61-
17. Example. Consider again the map f given in example 16. First
t
Next
Zf
= {3,
f
=
5, 12, 14, 17, 18} and p
o = 1ff
W
m1 = 1
m2 = 1
wI = f(3) = 9
o =8
W
z1 = 3
z2
= 5
z3
= 12
Z4
= 14
z5 = 17
z6
= 18
8 13 9 4 6 20 16 10
w2
w3
m3 = 3
m4 = 1
mS
m6
= card
Zf
= 6.
We then have
13 9 4 6 20 16 10
= f(5) = 8
= f3(12) f2(12)
= f(14) = 1
W = f3(17) f2(17)
s
w6 = f3(18) f2(18)
f(12)
=4
1 19
f(17)
=8
=9
11 2
w4
=3
=3
f(18)
15 7
Thus
t
o ...
= W
w6
=8
The notion of
13 946 20 16 10 9 8 4 1 19 1 8 11 2 9 15 7 .
Z - factoPization we now introduce is fundamental for
the contruction of the inverse of the map
f ~
f.
A letter xi (1 s i s n)
w = xl x2 ••. xn is said to be muZtipZe in w if either
or there exists j with 1 S j < i and x. = x. •
of a word
J
18.
Proposition. Any bJOi>d
(w ' ~1' ••• , wp ) (p ~ 0),
O
i
=1
1
2 .•. xn admits a unique factoPization
w = xl x
caZZed its
= 0,
Z - faator'ization~
having the fo1,-
1, .. , P the fir'st letter' of wk is
the only letter' of wk that is muZtipZe in Wo wI'" wk •
Let x.1 , x.1 , •.. , x.1 (1 = i O < ••• < i p S n) be the sequence of the
lowing pr'oper'ty
0
(M)
for' any k
1
muZtipZe "letter's of w.
p
Then
,
-62-
o = Xl
(47)
W
•••
X.
I'
~l-
WI
= x.~I
• •. x.
~2-
I' ••• ,
W
P
= x.1
p
and there is no repeated 'letter- in eaah of the UJords wo' wI' .•• , wp •
Proof.
(w O' WI' •.• , wp) defined by (47) satisfies condiOn the other hand, if w had repeated letters, there would be
k
The sequence
OM).
tion
a letter in wk
other than the first one to be multiple in wk '
in Wo WI". Wk.
perty
a fortiori
Clearly, there can be only one factorization having pro-
Q.E.D.
(M) •
As is now shown, this proposition will imply that
tion of F
f ~ f
is a permuta-
Consider again f
n
in (44) and (45).
in Fn and keep the notations introduced
Further, define for any k = 1, ••• , P the word
(48)
so that
19.
Proposition.
in (44) and (45), is the
more3
(resp.
o wi
W
(W O' WI' ••• , Wp ) UJhose terms are defined
Z - faatorization of f = Wo WI' •• Wp ' Further--
The sequenae
... wp
(resp.
o WI
W
wp )
is a pearrangement of 1 2 ••• n
f(l) f(2) •.. fen))
m
Proof.
From the definition of wk given in (45) the letter f k(Zk) is
multiple in Wo ... wk. Moreover, the words
mk-l
for
f
(zk) •.. f(zk) have no letter in common.
~ ~ i
> j
~ l , the element fj(Zk) would be recurrent, and occur in
wO .
TIlis would contradict the definition of mk • Hence,
f~(Zk)
is
-63-
the only letter of wk which is multiple in
o ...
W
wk'
The sequence
(wO' wI' ..• , wp ) is then the Z - factorization of f = Wo wI'" wp
Next consider the juxtaposition product Wo wi ... w
where
p'
p
wi, ... , w
are defined in (48). .From the first part of the proof it
follows that the product has no repeated letter.
range of f
an element
contains
If x belongs to the
and is non-recurrent, there exists, according to lemma 15,
...n-1
n-2
zk of Zf such that the word ~
(zk) f
(zk) ..• f(zk)
x.
If zk
is the smallest element of Zf having this prom -1
perty, then x is necessarily a letter of f k
Since the word
elements of
o wi
W
[n],
p
..• w
(zk) .•. f(zk) zk
= wk
has no repeated letter and contains all the
it is a rearrangement of 1 2
(45) and (48) we also conclude that
o wI'"
n.
Hence, from
wp is a rearrangement of
Q.E.D.
W
f(l) f(2) ... fen) •
Fn we form the flow ref) = [i(l) ;(2) ::: ~(n)]'
is a permutation, identified with the word fell f(2)
fen) ,
If f
Waen
f
then
ref)
belongs to
is the circuit defined in (35) with
X = [n] • We let
Gn be
the set of all nn flows of the form
1
[
where
g
f
Fn
Another bijection
= gel)
runs over
g(2) ... g{n)
For instance, with n
[~~~:] [~~].
2
fell f(2)
Clearly r
is a bijeation of Fn onto Gn
~ of F
n onto Gn is defined as follows. If
is a permutation, ~(g) has been defined in (39).
=9
and g
= 316872495
we have
~(g) = [;~] [~]
When g is not a permutation we form the Z-factorization
-64-
(wo' wI' .•• , wp ) of the word gel) g(2) •.• g(n).
repeated letters, there is a unique rearrangement
As
wb
II (wO) •
1
[:~
(with
II
as in (39)).
o has no
W
o defined
W
of
Next. we observe that if
(p+ 1)
the numbep of faatops of the Z-faatopization of gel) g(2) •.• g(n),
the numbep of el.ements of en]
is equal. to p.
are in gel) g(2) ... g(n)
Let then
integers of
then
whiah do not oacup in gel) g(2) ••• g(n)
before each mUltiple letter.
as many missing elements of
(zl' z2' ••. , zp)
Acordingly, there
en]
as mUltiple
be the increasing sequence of the
which do not occur in gel) g(2) ..• g(n).
Also write
is the first letter of wk (k = 1, 2, •.• , p) ,
and let w = vk zk (k = 1, 2, ... , p). The word
wi ... w is a
since we have replaced each multiple letter
rearrangement of 1 2 •.. n
Wk
= Yk
en]
is
Indeed, the Z-factorization of g is obtained by cutting
the word gel) g(2) ..• g(n)
letters.
by
vk '
where Yk
wb
k
of
o wI'"
W
p by a missing element.
w
(49)
Thus
lI(g) •
6 maps
Fn
Ti(l) ~(2)
into
Gn
[:~1 [:~]
p
We then define
\
..,[::]
Let us prove that
6
is bijective.
~(n)1.
Let wo' wI' ... , wp ,
the words defined in (44), (45) and (48). If (i l , · · · , i r )
flow
:::
.
hI
ts o
f f
R
lng
sequence 0 f tee
emen
'
be written as
II(WO) ·
I:~]
wi, ... , wp be
the circuit .[f(i
i 1 ) •••
...
l
for some rearrangement
wb
Take a
is the increasmay
-65-
Since
(48)
and
(50)
it follows from the second part of proposition 19 that
1
2
[. f(l) f(2)
(51)
000
0 ••
n
]
=
fen)
o] r:i ]
r:tot
000
1
r:~]'
tp
Now the first part of the same proposition says that
the Z-factorization of
o wI
W
""
••• w = f .
P
(w O' wI' ••. , wp) is
On the other hand, (zl' z2'
is the increasing sequence of the elements of
o wI'" wp ' Furthermore wk
wk = vk zk (k = 1, 2, ••• , p).
W
... ,
[n]
which do not occur in
and wk are of the form wk = Yk vk '
We then conclude from the definition of 6
given in (49) that
This shows that
G ,
n
6 is surjective.
Hence 6 is a bijection of Fn onto
as the two sets have the same cardinality. Finally, with 6- 1 denoting
the inverse of 6,
The map f
20.
f is then a bijection of
+
Example.
f
we have
=
n onto itself.
F
Let us take again example 17.
We had
8 13 9 4 6 20 16 10 9 8 4 1 19 1 8 11 2 9 15 7 •
zp.1..
-66-
Its Z-factorization is
(8 13 9 4 6 20 16 10, 9, 8, 4 1 19, 1, S 11 2, 9 15 7)
and the list of the missing integers of
17, IS).
... ,
20}
is
(3, 5, 12, 14,
As in (39) we have
6(8 13 9 4 6 20 16 10)
fa
= .~
9 4 6 13 16 10 20]
13 9 4 6 20 16 10 •
Next by associating each missing element
wk of
{l, 2,
zk with the corresponding factor
f we obtain
=
fs
La
9 4 6 13 16 10 20-J r39}·
13 9 4 ;6 20 16 10 U
rs]·
La [14 191 12]
19
[14]
1
[11a 1121721[159 15718}7
Rearranging the columns we obtain
,.
[1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20J
6(f) =4 11 9 9 a 4 15 a 13 16 a 19 6 1 9 20 2 7 1 10
and so
21.
f
= r- 16(f)
Remark.
is the bottom row of 6(£)
Formulas (41) and (59) are similar.
Each exhibits the exist-
ence of a fundamental transformation in a particular setting.
In each case
we have made use of the properties of the flow monoid to construct the fundamental transformation.
In the mu1tipermutation case we dealt with cipauits
only, that is flows of the form
.[:']
with .w I
However, the one-column matrices within·
with each other, since w'
semi-group case the flows
[:'1.
a rearrangement of w.
did not necessarily commute
may have repeated letters.
r
w
~ ']
w
a rearrangeQent of 1 2 ..• n.
In the symmetric
are not necessarily circuits, but w'
is
Accordingly, all the one-column matrices
-67-
[:'J.
of
commute with each other.
subsets of the ftow monoid.
into the flow monoid
problem arises:
F(X)
We have then considered two particular
In each case an injection
has been defined.
rand 6 of
x*
Accordingly, the following
find a natural construction which includes the fundamental
transformation for
multipermutati~.and for
functions as special cases.
As expected, there is also a result about exceedances and descents for
the symmetric semigroup case.
at formulas (50) and (51).
w
...
w
rf 00']
.
The result will become apparent if we look
1
• •. n
]
[ f(l)
... fen)
The flow
is the product of
and flows of the form
(53)
(k
= 1,
.•. , p).
Moreover the word
f (n)
£(1) f(2)
is the juxtaposition.
product
It follows from proposition 3 that
o contains a factor yx with x
W
if and only if x is recurrent and x
<
y
= f(x).
<
y
Suppose now x non-
It is clear from (53) that fex) = y and x € fern]) if and
mk-l
only if yx occurs in some factor wk = ff
(zk) ... ff(zk) f(zk)' say
fi+l(Zk) fiezk) = yx, and fi(Zk) non-multiple in f(l) f(2) •.• fen) •
recurrent.
The following notations are then to be introduced.
(x,y)
of distinct integers of
[n]
equal to 1 if either x is recurrent for
x is non-recurrent for
f,
f(x)
in
and f
=y
n we let vx,y (f) be
f(x) = y and x < y, or
F
f
and x
For any ordered pair
€
f([n]).
In the other cases
-68-
we let vx,y (f) be equal to O. Next ~ x,y (f) = 1 if either x < y ,
f(l) f(2)
fen) = vyxv' and vyx has no repeated letters, or
f(l) f(2)
= vyxv'
fen)
repeated letters.
vy has no letter equal to x but does have
x,y (f) = O. We conclude from this discussion
(x,y) of distinct integers of [n] and any f
Otherwise
that for any ordered pair
in Fn
,
~
we have
(54)
Let us take up again examples 17 and 20 .. We have vX,y(f)
= ~X,y(f) = 1
for the following pairs
(9,13), (4,9), (16,20), (10,16), (1,4), (19,1), (11,8), (2,11), (15,9), (7,15) .
We can also say that an exaeedanae of f
x < f(x)
and
is a pair
x an element of the range
(x,y)
factorized as
=
r
v
(f)
x<y x,y
(x, f(x))
of f.
with
A desoent of
f(l) f(2) ... fen)
with x non-occurring in v.
ances (resp. descents) is the~ Ef
~~reover
fern])
with x < y so that the word
vyxv'
is a pair
f
can be
The number of exceed-
(resp.
Df
= x<y
r;x,y (f)).
(54) implies that
(55)
Ef
-=
,.
Df.
In examples 17 and 21 the only exceedances (resp. descents) of f
f)
(resp.
Thus
Ef
are
= Df = 7
(9,13), (4,9), (16,20), (10,16), (1,4), (2,11), (7,15) .
.
The fundamental transformation appears to be most useful when we have
to evaluate the cardinalities of special classes of functions.
a few examples.
We list here
.. 69-
22.
Counting Classes of Functions.
1.
A map f
of
[n]
into itself is said to
"1"
beacyc~~c
Le. if all its recurrent elements are fixed points.
to such a map is then a ZabeZed rooted forest.
with length r
with length n - r - I
w xw'
having its letters in
A
f-+f
of this form is the image under
0
fixed points.
If f
[n]
is
W
.
acyclic and has
o increasing
and w' any. word
with
x a letter equal to a letter of
= fn-1
The graph associated
= wOxw '
f is of the form f
fixed points 1 then
r
1f f n
o
W
Conversely, any word
of an ·acyclic map with
r
A ) be the set of acyclic maps (re:sp.
n1r
fixed points). We then have
Let
A
n
acyclic maps having r
(resp.
rn n-r-l .
In particular 1 the generating polynomial of the number of fixed points over
A
n
is
I
lSrsn
xr Card A
n,r
= x(x+n)n-l
.
Finally
n-r-l
n-l
= (n+l)
.
Card A I
[nJ rn
n - lSrsn r
2.
The connected maps are the maps having only one cycle 1 possibly
reduced to a fixed point.
Let
Bn
(resp.
nected maps (resp. connected maps with
in
Bn,r 1 the word
maximum letter.
n - r - I
x
r
) be the set of all conn,r
recurrent elements). If f is
B
f is of the form woxw' with
equal to a letter of
having its letters in
[n] •
W
o
Hence
and
w'
o starting with its
W
any word with length
-70-
Card Bn,r
= (~)
n
(r-l)! rn - r - l
and
Card B
n
=
r [n-l)
Osrsn-l r
nn-l-rr!.
The generating function of the signless Lah numbers
3.
is given by
(see Riordan (1958), pp. 43-44)
(n > 0).
Let
Ln denote the set of all acyclic maps f such that
is at most 1 if x is non-recurrent and at most 2 if x
card f-ICx)
is recurrent.
The graph associated to such a map is a "forest of trees
without leaves".
Using the fundamental transformation it is straight-forward
Ln (x) is the generating polynomial of
the number of fixed points over the set Ln
to show that the polynomial
4.
As was shown by
Fran~on
(1974) the fundamental transformation pro-
vides an elegant way of establishing the Abel identities.
We give here only
one example, namely we prove the identity
(56)
where xl' x ' a are three commuting variables.
2
Let us consider the set
An +2,1,2 of all maps of [n+2] into itself having 1 and 2 among
their fixed points. If f is such a map, then f is of the form 1 2 w'
where
Let
Wi
b(f)
is any word with length n whose letters are in
= card
f-l(l) , cef)
= card
[n+2].
f- l (2) , d(f) = card f- 1 ([3,n+2]),
a~ld
-71-
P(f)=X~(f)x~(f)ad(f). Then the generating polynomial P = X{p(f):f€An+2 ,1,Z}
is obtained from the formal sum L{f:f€An+ 2 , 1 , 2} by replacing each occurrence
of 1 (resp. 2, any i > 2) by xl (resp. x2 ' a)
Accordingly
n
P = Xl x 2 (x l +x 2+na) . A map of An+2,1,2 consists of an acyclic eonneeted map fl whose root is 1, and a map fll having 2 among
its fixed points.
of
[3, n+2]
Denote by X(fl)
with k elements
(0
the domain of fl • For any subset
~
k
~
n)
X
we have
L p(f) = LP(f') p(fll) = Xp(f') LP(fll)
X(f')={l}uX
As
there are
[~)
subsets
X of
[3, n+2]
with cardinality k,
formula
(56) is obtained by summation on k.
5.
In the last example we give the expression for a bivariate generat-
ing function over A
n
Formula (57) below was obtained by Riordan (1973)
who made use of traditional methods of differential calculus.
Here we
apply the techniques of the fundamental transformation .
For any acyclic map
f
in An
let
number of fixed points (resp. of elements
The proof goes as follows.
An
a(f)
(resp.
i
with f(i)
b(f)) denote the
= 1).
As above, let us associate to any map
the connected acyclic map g in An+ l
defined by
Then
f
in
-72-
g(x)
= f(x)
if 1 S x
nand
f(x)
~
= n+l
if 1 S x s nand
f(x)
=x
= n+l
if x
~
= n+1
x
•
Denote by A
l l the set of all connected acyclic maps over [n+l]
n+ ,n+
the single fixed point n + 1 . lfuen g is in An +1 ,n+ l ' then
g
= (n+1)
(n+l) w ,
belong to
where w is a word with length
[n+l] .
are three kinds of words
(ii)
of
z
~
and make appropriate substitutions.
g to consider
(n+l) (n+l) (n+l) v or
in v
1
(iii)
1, n + 1 and
(i) g = (n+l) (n+l) 1
(n+l) (n+l) v (n+l) 1
(n+l) (n+l) z v or
(n+l)
and so must be replaced by 1.
replaced by xy,
[2,n] _.
(n+l)
with
VI
refers to the fixed point
gel)
in
(n+l)
f(l)
= 1.
Finally, any letter of
(n+l) ,
1,
VI
Then
is
or any
Hence
If g
= n+l
= (n+l)
and so
of g must then be replaced by xy.
rence of
(n+l) (n+l) v z 1
1 according as it is equal to
Consider case (ii).
then
with no occurrence
since it corresponds to the fixed point
replaced by x,y or
1 in v,
VI
In case (i) the second letter is
y is substituted for the third letter 1.
integer of
VI
There
no occurrence of 1 in v .
In each case the first letter
of g,
whose letters
To obtain our generating polynomial we may consider the
r{~:g€An+l,n+l}'
formal sum
(n-l),
with
(n+1) (n+l) v with no occurrence of
fell
= 1.
The second letter
(n+l)
In the same manner, the third occur-
(n+l) (n+1) v (n+1) 1
VI
is to be replaced by xy.
-73-
Next consider the map defined by
(n+l) (n+l) (n+l) v
(n+l) (n+l) (n+l) v
+
or
(n+l) (n+l) v (n+l) 1 v'
+
(n+l) (n+l) (n+l) v 1 v' •
It maps in a one-to-one manner the set of the words of case (ii) onto the set
of all words with the form
with length
in the letters
(n-2)
(n+l) (n+l) (n+l) v"
If
v"
=V
l.v'
(n+l) (n+l) (n+l)
l, 2,
Vii,
...,
where v"
is any word
n+l • The preimage of
is the same word if v" has not letter equal to
with no letter equal to
1
in v,
the preimage is
Henc~to obtain I{xa(f)yb(f): case (ii)} ,
(n+l) (n+l) v (n+l) 1 v' .
1 •
we
have to consider the formal sum
I (n+l) (n+l) (n+l). v" ,
replace
n+l
(n+l) (n+l) (n+l)
(resp.
I,
by l.x.xy,
any integer of
[2,nJ)
and, as above, any occurrence of
in v" by x
Case (iii) is treated in an analogous manner.
onto itself, and
(n+l) (n+l) v z 1 v'
onto
(resp. y,l).
We map
Thus
(n+l) (n+l) z v
(n+l) (n+l) z v 1 v' • Then,
we consider the formal sum
I (n+l) (n+l) z v" ,
and replace
by l .
(n+l) (n+l)
Next, in
by l.x,
since the second
zv" we replace any occurrence of
(n+l)
(n+l)
is not followed
(resp.
l,
any
-74-
integer of
[2,n])
by x
(resp.
y,l).
As
z
~
1, n + 1,
we obtain
I{xa(f)yb(f): case (iii)} = x(n-l) (x+y+n_l)n-2 .
Formula (57) is obtained by swmnation of cases (i), (ii) and (iii).
Notice
that
~
l{X
a(f) yb(f) :f€A ,f(l)=l}
n
(x+y+n-l) n-2 .
= x(y 2+xy)
With some patience and effort we should be able to give an expression for
~{ xa(f) Yb(f) zc(f)'f
. €A
l
where
c(f)
= card
1
f- (2).
n
}
,
Let us await discovery.
The fundamental trans-
formation in the symmetric semi-group case was first published in FoataFuchs (1970).
-75-
7.
A Probabilistic Lemma and Some Applications.
Most of the above results on rearrangements may be stated in probabi·
listie terms.
It is the purpose of this section to give a simple lemma
which will make possible the transcription into probabilistic language.
Of course, there is no mystery behind this lemma.
It can even be considered
selfevident. We only mention it here for the sake of completeness.
In this section words with length n whose letters are real numbers
are identified with elements of the Euclidean space Rn • Let U and V
be two (m~asurablel) real functions defined on Rn • Suppose that there
exists a permutation ~ of Rn with the following two properties
(i)
U(w) = V~(w)
for any win Rn ;
(58)
(ii)
t(w)
is always a rearrangement of w .
For any w in Rn we let Cw be the class of all rearrangements of w.
The restriction of ~ to any class Cw (which is necessarily finite) is
a permutation of Cw • For any real y there are then in Cw as many
elements w' with U(w') = y, as elements wIt with V(w") = y. Thus
card C n U-1 (y)
w
and for any subset
= card
Cw n V· 1 (y) ,
B of R
(59)
In probabilistic notation I{Ue:B} is the indicator of the "event" {Ue:B}.
For any w' in Rn we then have
-76-
I{UEB}(w') = 1 if U(w') E B
=0
otherwise
Using this notation (59) can be rewritten as
(60)
Now for any w = Xl x 2 ... x n in Rn and any permutation a
of 1 2 •.• n we let aw = x. x. . .. x.
Finally let
J.
J.
J.
l
be n (n > 0)
2
real random varib1es.
=i1
i2
i
n
n
We say that they are exohangeabZe (or
in symmetpia dependenae) if the distribution P of the vector X = (X l ,X 2, .•. ,XJ.
is invariant under permutation of the subscripts
dom vector
X = (X1 ,X2, ••. ,Xn )
then u = UeX and v
= VOX
1, 2, ••• , n.
If the ran-
(O,A,p),
is defined on the probability space
are real random variables also defined on
(O,A,p) .
23.
n satisfying conditions
Lemma. If ~ is a pe'llmutation of
R
(i)
and
(ii) of (58), then the random vaPiabZes u and v defined on (n,A,p)
(resp. U and V defined on (Rn , sn, P)) have the same distpibution.
Proof.
It suffices to prove the "resp.1I part. Let S be the a-field of
all symmetric Borel subsets of Rn , that is to say, Borel subsets S which
contain the entire rearrangement class
f
Cw as soon as
be a P-integrable function (defined on
trization, Le.
(Rn , Bn , P)
w is in S.
and
f
Let
O its symme-
-77-
(61)
The function
f O is S-measurable.
Moreover, as
Xl' X2 , ..• , X
n
are
exchangeable, we have
I
= Is
few) dP(w)
S
for any S in S.
with respect to S,
.
Rn we t h en have
ln
fo(w) dP(w)
Consequently f
i.e.
is the conditional expectation of f
O
f O = E[fIS]. For any real Borel set Band w
P{UeBIS}Cw)
= E[I{UeB1IS](w) =
= (l/card
Cw) L{I{UeB}(w')
w' e C } •
w
Hence, identity (60) yields
P{UeBIS}
= P{VeSIS}
a.s.,
and
P{U€l3}
Thus
U and
= JP{UeB IS}dP = JP{VeB ISldP = P{VeB}
•
Q.E.D.
V have the same distribution.
Let us use lelnma 23 to give a probabilistic corollary of theorem 14.
Let there be given a Borel function b: R2 + R satisfying
(62)
b(x,y)
With any w = xl x 2 ... xn
in
=0
if x
~
Y .
n
R associate its non-decreasing rearrangement
here denoted by xCI) x(2) ..• x(n)
to match with the usual notation for
-78-
aPder statistias.
Next let
v(w) = <....
I b(x(.),x.)
1. 1.
1 _1.;:,n
(63)
and
~(w)
=
I
l~i~n-l
b(x.,x·+ l )
1. 1.
We may write
v(w) =
I b(x,y)'card{i:lsisn, x(i)=x, xi=y}
x,y
But card{ .•. } is precisely vx,y (w) defined in theorem 14. Hence
v(w) = I b(x,y) vx,y (w) which is equal to I b(x,y) vx,y (w) because of
x<y
x,y
assumption (62) . In the same manner
~(w) =
I b(x,y) ~ x,y (w) = x<y
I b(x,y) ~ x, yew) •
x,y
Now theorem 14 implies that
V(w)
As
~
24.
= l;; (~)
•
is a rearrangement of w we may then apply lemma 23.
We then obtain
Corol1ary of Theorem 14. Let Xl' X2, •.• , Xn be n (n>O) exahange-
able reaZ ra,uiom variables and
(XCI)' X(2 )' ••• , X(n»
be the order statistias
of (Xl' X2, •.. , Xn) • Further, Zet b: R2 + R be any Borel funation ~ith
the property that b(x,y) = 0 if x <?: y. Then the two statistias
I
1<·...
-1.~n
I b(X.,
X1.'+l) have the same distribution.
1<'<
1.
_1._n -1
Note that the above corollary holds for any sequence of exchangeable
b(X(.),x.)
1.
1.
and
random variables.
crete or singular.
In particular the distribution of the
X.1. 's may be dis-
-79-
We now turn our attention to a famous question in Probability Theory
that turned out to be of a combinatorial nature as was shown by Sparre
Andersen.
In the next section we shall discuss the algebraic aspect of
the so-called Spapre Andepson equivaZenae ppinaipZe.
Here we state its
probabilistic version.
Let
variables.
Xl' X2, ••• , Xn
So = 0 and for any n > 0 denote by 5n the n-th
(~l)n>O
be a sequence of exchangeable random
This means that for every n > 0 the variables
are exchangeable.
Let
paPtiaZ sum 5n = Xl +
+ X
The index of the fipst maximum of the
n
, Sn) is the random variable Ln defined by
sequence (SO' 51'
...
for any k
= 0,
1, ••. , n.
On the other hand, the numbep of positive paP-
tiaZ sums in the sequence
is the random variable
IT
n
defined by
{IT =k} = {exactly k partial sums Sj are
n
positive among SO' 51' , , 5 }
n
..
(k
= O,l, .. .,n) .
Theorem.
abZes
L
n
(Equivalence principle).
Fop any n > 0 the pandom vaPi-
and IIn have the sOJ'r.e distp-i.bution.
The distribution of
bution of each
X.~
Ln (or IT ) is knO\VTI explicitly when the distrin
is symmetric (i.e. P{X.>x}
= P{X.<-x}
for any x > 0)
1
~
and absolutely continuous.
In this case we have
(see e.g. Feller (1966), vol. 2, pp. 398-401)
-80-
P{L =k} = P{TI =k} = (2kJ (2n-2k) 2- 2n
n
n
n
n-k
(O~k~n),
leading to the famous are-sine Zaw
for any a
with
0
< a <
However the equivalence principle holds for
I.
any sequence of exchangeable random variables.
For any vector w = xl x2 •.. xn in Rn (n>O)
tial sums
Then we define
L(w)
as the subscript of the first maximum in
and
the sequence
TI(w)
pew)
as the number of positive terms in
(sO(w), sl(w), •.. ,sn(w)).
suffices to construct a bijection
that
In order to prove theorem 25 it
of Rn onto itself with the property.
p
is a rearrangement of wand
TI (w) = Lp (w)
(64)
for any w in Rn .
Such a bijection
p was first found by Ian Richards,
as quoted for the first time in Baxter (1957).
p (e)
=e
and for any word w = xl x
(65)
we can define the par-
p(wx)
if e
2
It is defined as follows
is the empty word,
... xn in
= p(w)x
if sn+l(wx)
= xp(w)
otherwise.
n
R
= xl
(n~O)
and
+ ••• +
x in
xn + x
~
R
0
-81-
An alternate definition is:
denote by
(i 11 i 21 .•. 1i p )
of the subscripts
(p+q = n)
i
w = xl x 2 •.• xn be a non-empty word and
(resp. (j11 j2 1"' 1 jq) the increasing sequence
le~
\\lith the property that
xl + ... + xi
>
0
~
(resp.
0)
Then
(66)
p (w) =
Clearly pew)
is a rearrangement of w.
Thus
prove by induction on the word length that
sn(p(w))
= Sn(w).
Let us
is a bijeation of any
P
reQXIrangement a"lass Cw onto itseZf. There is nothing to prove if w is
empty. Let v belong to Cw with w with length n > O. If sn(v) ~ 0
(resp.
p(v")
= v'
=v
v'x
p
> 0)
•
we write v
As
(resp.
s
n
(v"x)
p(v"x)
= v'x
(resp.
(resp.
~ 0
xv').
There is a word v" with
we have
> 0),
= xp(v") = xv' =v)
p (v"x) = p (v") x
1 by definition of p.
=
Hence
is bijective.
The proof of property (66) goes as follows.
if w is empty.
(n~O)
Let us proceed by induction.
and x be a real letter.
We first note that
s.1 (wx)
II (wx)
(67)
When
sn+l(wx)
~
Let w have length n
II(w)
= Lp(w)
1, "'1 n •
Hence
Assume by induction that
= s.1 (w)
= II(w)
= II(W)
There is nothing to prove
if
+ 1
for any
s n+ l(wx)
if
5
= 01
i
~
0
n+ l(wx)
0 1 the first maximum in
> 0 .
(so(wx)1 ..• ,sn+l(WX))
is also
......
the first maximum in
and
in
On the other
the."first maximum in (so (w)
is
l ' ••
1Sn (W))
is
Sp+l(XW) • Thus
han~
if
s n+ l(xw)
the first"" maximum
> 0 ,
-82-
L(wx)
= L(w)
if
(68)
L(xw) = L(w) + 1
5
~
n+ 1(wx)
0
if sn+l(xw) > 0
Relation (66) is then a consequence of the definition of p,
(67) and
(68) •
Example. Consider the word w = xl x2 x3 x 4 Xs with xl
26.
= -1
,
, x = 0 , x = -2 , Xs = 1. In the Euchidean plane draw the poly3
4
gonal line joining (0,0), (l,x ) , (2,x 1+x 2) , ... , (S,xl+,,,+x S) ' We
l
obtain
x2
=3
/.
4
Then
i3
TI(w)
= 3,
the subscripts
= S. Let jl = 1 ,
jz
=4
i
----~
5
with si (w)
. Then the word
>
0 being
pew)
\
_--L.._--'-_-"
4
5
- .. '---)
is
il
=2
, i
Z
=3
,
-83-
The first maximum of
(sO(p(w)) , •.• , sS(p(w)))
is
s3(P(w)) • Accordingly
L(p(w))= 3 •
27.
Table.
It seems interesting to give the description of the bijection
p restricted to a rearrangement class.
with w =
w
-2
-2
-2
-1
-1
-1
1
1
1
1
1
1
1
-1
1
1
1 -2
1 1
-2 -1
-2 1
-1 -2
-1 1
1 -2
1 -1
-1
1
1
-2
~2,
Here we consider the class
Cw
-1, 1, 1 •
51 (w)
SZ(w)
s3(w)
s4 (w)
II(p)=Lp(w)
L(w)
-2
-3
-2
0
0
-2
-1
-2
-1
-2
0
0
-z
-2
0
0
0
-2
-2
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
0
0
0
0
1
3
1
1
1
1
1
2
2
-1
-1
1
1
1
1
1
1
-1
1 -1
1 1 -1
-2 1 1
1 -2 1
1
1
-1
1
1
-1
-1
-3
-1
0
-2
-2
1
-1
1
0
1
-1
-2
-1
1
-2
-1
-2
1
-1
0
1
1
1
0
-2
1
1
0
2
2
0
1
1
1
2
2
3
.'p (w)
1
1
1
1 -2
-2 -1 1
-2 1 -1
-1
-1 -2
1
1 -1 -2
1 -2 -1
1 1 -2
-84-
8.
The Equivalence Principle of Sparre Andersen. An Algebraic Study.
First, we restate the equivalence principle given in the previous
section.
Instead of starting with the set R of real numbers, we consider
an abstract set
X and form the free monoid
we suppose that
s
X*
is a fixed homomorphism of
generated by X.
x*
into the
Next,
additive
w = Xl x 2 ... xn be a non-empty word of X*. We
then have sew) = s(x l ) + x(x 2) +
+ s(x ).
If w is the empty word,
n
then sew) = O. The Zength of a word w will be denoted by lew)
The
group of R.
Let
partiaZ sums of a word w will be the real numbers
any left factor of w.
sew') ,
where w'
is
Clearly, the two subsets
= {W€X*
p*
sew) > O} u {e}
and'
= {w€X*
Q*
sew)
~
O}
are submonoids with the following two properties
X*
(69)
The number
new)
= p*
p* n Q*
pp*
= P*\{e}.
e is a factor of any word w.
the subset of all words w with
left factor w'
= {e}
of positive partial sums is then the number of left fac-
tors of w which belong to
empty word
n Q*
of w.
We make the convention that the
Further we designate by A*
sew') < sew)
For any word in A*
for any proper (i.e.
= {W€X*:s(w')
~
w)
the maximum of the partial
sums if then reached at the last term. Let also
B*
~
0 for any left factor
w'
of w} •
-85-
Clearly,
A* .and B* are both submonoids of X*.
with length n
= p.
lea)
>
0 and suppose that
Clearly a belongs to A*
Let w be a word
=p
We write w = ab with
and b to
B* and there is no
L(w)
other factorization of w as a product of an element of A* by an element
of B*.
The subscript
L(w)
of the first maximum in the sequence of
partial sums is then the length of a
w with a
that
the~e
in A*
and b in B*
exists a bijection
ment class onto
p
in the unique factorization
of x* onto
Instead of using the homomorphism s
have started with a pair
Such a pair is
~aid
(P*,Q*)
itself~
= Lp(w)
(A*,B*)
~ea~~ange­
mapping any
holds identicalZy.
to define. IT we could as well
of monoids satisfying relations (69).
to be a partition of X*.
we need to have a pair
For the definition of
pair
in A*
Such a pair is called a factorization of X*: To know what
B*
(A*,B*)
L
of monoids having the following property:
any non-empty word w has a unique factorization w = ab with a
and b in
of
Then the equivalence principle says
and such that new)
itseZf~
ab
to associate with
(P*,Q*)
we have to discuss first the
relationship between partitions and factorizations of x* .
If 11 and M'
of all words
ww'
are two subsets of X*,
with w in M and w'
denote the submcnoid generated by H,
particular,
S
= (M\{e})
M.
Xx*
= X*\{e}.
\ (M\{e})2
The basis
S is
and
in M' • Further M* will
MM*
the set M*\{e}.
of M is the least subset of X*
In
that generates
still characterized by the following property:
is a proper subset of S,
but not M.
the set
If M itself is a submonoid, the basis
submonoid N of x* which contains S,
S'
we denote by Mr-,,1'
also contains
there is a submonoid M'
of
M;
any
while if S'
X* which contains
For instance the basis A of the submonoid
-86-
A*
= {WEX*:S(W')<s(w)
for any proper left factor W'
of w}
is
A = {w: O<s(w)
and
The reverse image
w
= Xm
for any proper left factor
W'
of w} .
W = Xl Xz •.. xm is the word
'"
By convention '"e = e • Let A*
(resp. '"B*) be the
w of a non-empty word
.
Xz Xl
s(w')sO
set of all words
W with W in A*
a factorization of
x*
(resp.
if and only if
.
B*)
The pair
('8* ,A*)
(A*,B*). is a factorization of
is
X* .
Accordingly, all our following statements have a symmetric counterpart by
exchanging
28.
p*
Q*, A* and B*,
and
Lemma. Let
(A*,B*)
and left and right.
be a factorization of X*.
Then
A*
(resp.
B*)
is a right (resp. left) prefix monoid., i.e.
a€A*,
aw€A*
implies w€A*
b€B*,
wb€B*
implies WEB* •
and
Proof. Take a and w as above.
If w = e,
If w is non-empty, we have w = a'b'
(a a')b'
b'
29.
= e,
= aw€A*.
with
the conclusion is trivial.
a'EA*
and b'EB*.
Hence
Because of the unicity of the factorization this implies
that is w
= a'EA*
.Q.E.D.
•
Lemma. Let A be the basis of A*. Then every word w has wxactly
one factorization of the form w
in A and c in
X*\AX*.
= al
aZ
-87-
Proof. We proceed by induction on the largest m ;::;
If
m=0
there is nothing to prove.
with aI' a' in
1
other, say a 1
prefix.
u
As
= e,
30.
A.
= alu
al,ai
that is
If
such that
w€AmX*
.
o , suppose w = alv = a'v'
1
and a' must be a left factor of the
1
l
But then u is in A* since A* is
One of a
, u€X*
.
belong to the basis
= ai
al
m>
0
A of A*
we conclude that
Q.E.D.
.
Proposition. A necessary and sufficient condition that (A*,B*) be a
factorization of x* is that A* and B* be submonoids with the three
properties
(i)
A* n B*
= {e}
;
(it): A*· is right prefix and
(iii)
= A*B*
x*
Zeft prefix;
B*
.
Proof. The necessity of these conditions follows from the definition for
and (ii) and from lemma 28 for (ii).
(i)
To prove that they are sufficient we
have only to show that under conditions (i), (ii) and (iii) any relation
ab = alb' (a, a'€A*, b, bl€B*)
ab
is
= alb'
a
a
= a'
we may assume for instance that
= a'u.
Then b'
and also u€B*, because
that u
implies
=e
= ub
B*
As
A*
a'
Indeed, if
is a factor of a,
that
is right prefix, we have u€A*
is left prefix.
By condition (i) we conclude
Q.E.D.
.
31. Lemma. Let (A*,B*) be a factorization.
(i)
, b = b'.
every right factor of a word of
A~
Then
is in A*
-88-
every proper Zeft faator of a word of A is in B*;
Bm An c''A U A2 u ••• u An u B u B2 u ••• u Bm (m,n;?!:.
0)
(ii)
(1' l' l' )
Proof. Consider a word a
= vweA*.
or w is the empty word.
If v and ware different from e,
v
= alb'
Clearly w is in A*
we have
, w = a"b" , a', a"eA* , b', b"eB* .
Again a' b t a" = a I b 1 with
a = ae = al(b1b"). Because of the unicity
al
in A*, b l in B*. Thus
of the factorization we conclude that
proves (i).
a
if either v
bIb"
=e
and so w = a"eA*.
= a2
b eA*B*
2'
In similar fashion, we have b'a"b"
= ae = (a'aZ)b Z
we again conclude that
b2
=e
As
This
and from
w;t e and b"
=e
Hence az;t e because b Z = e. Thus, a = a'a Z belongs
to the basis A only if at = e, that is, only if v = b'eB*. As the
then a";t e
empty left factor of a also belongs to
Now let
=e
or as
admits
Then ba
we have then proved (ii).
= asb s
' aSeA* , b3eB*. From (i)
it follows that either bS = e, or b 3 is not a right factor of as'
that is, b admits a as a proper right factor. Symmetrically, either
S
as
aeA and beB.
B*,
b as a proper left factor.
Thus, one of the words
Suppose for instance bS = e we can write as = a'a"
By the symmetric version of (i) and ba = a S = a'a"
with a' e.A and a"eA*
we see that b· must be a proper left factor of a', that is, at = bu
as,b S must be empty.
with u;t e and ueA* by (i).
As
aeA,
it follows from (ii) that
case is excluded, we have a
proves
BA
HO\'lever ba = bua" shows that
c
A u B.
= u.
ueB or u
Hence a"
=e
= a.
a = ua" .
Since the first
and ba = ateA.
This
Condition (iii) follows by induction on m and n.
Q.E.D.
-89-
Two words
are said to be conjugate if they are cyclic rearrange-
w,w'
ments of each other, i.e. if one can find words
w = uv
32.
and w = vu .
Proposition.
Let
be a factorization of X*.
(A*,B*)
w has a conjugate in A*
Proof.
Let w = ab with
a
in A*
and
b in
show that none of the conjugates of a word
can write
a
any conjugate of a
A*
~
and uv
e
and u
in
B*.
m
with
has the form w'
= akEA.
Then a word
if and onZy if it has no conjugate in
31 the conjugate ba of w belongs to A*
v
u,v with the property that
or
a
B*.
By (iii) of lemma
B*.
Thus, it suffices to
in A*
belongs to
... , a
m
B*.
We
in A and so
= vak+ l •.. am a 1
By (i) and (ii) of lemma 31 we know that
If u
B* .
is empty, then w'EA*.
Otherwise,
v
is in
w' E AA*B*
Q.E.D.
so w'eB* •
We are now in a situation where we can relate factorization and partition of
X*.
To this effect let us define the right (resp. Zeft) associate
of a submonoid l1 of
X*
factors all belong to
M.
to be the set of words whose right (resp. left)
33. Lemma. The right associate of a submonoid p* is a right prefix
monoid whose basis A is such that
(X*\P*)* generated by the set
x*\p*
X*\AX*
is contained in the submonoid
-90-
Proof.
Let
a and a'
factor of aa'
is a right factor of a'
right factor of a.
associate A*
belong to the right associate of
As
p*
P*.
or a product ua'
Further A*
is right prefix since by
definition it satisfies the stronger condition that w'w€A*
to
A*.
with u a
is a monoid, this shows that its right
is also a monoid.
for any w, W'€X*.
Any right
Finally, if v is in
XX*\AX*
implies w€A*
it does not belong
Thus, it has a right factor v";z: e which belongs to
Let v = v'v".
Then
VI
e:X*\AX* •
X*\P*.
Induction on the length of v completes
Q.E.D.
the proof.
34. Theorem. A neaessa:r'Y and suffiaient aondition that
faatopi2ation of x* is that thepe exists a pa:r'tition
Buah that A*
(pesp.
Q*).
Proof.
Let
p*
(pesp.
(A*, B*)
B*)
be a
(A*,B*)
(P*,Q*)
of x*
is the pight (peep. Zeft) aSBoaiate of p*
be a factorization of
.
x*
as being the submonoid generated by X*\B* •
We define
w ;z: e
Let
Q*
= B*
be in
and
p*
.
X*\B*
W
As
Then w = wI w2 • •• wm with m > 0 and wI' w2 '
m in
we have wI = ab with ae:M* and
(A*,B*) is a factorization of x*
••• J
be:B*
.
Hence w e: {a}A*B*
and consequently that
(p* ,Q*)
is a partition of x*
is the left associate of Q*(= B*)
of
P*.
We have w i A*BB*
.
The fact that
= {e}
B*
follows from the symmetric version of
lemma 31 (i) • Again from lemma 31 (i) and
tained in the right associate of
p* n Q*
This proves that
and so w '- B* •
P*.
A*
c
p*
the set
A*
is con-
Let w belong to the right associate
since every word of A*BB*
has a right factor
-91-
in
B* and since
BB*
c
X*\P*.
Thus w€A*
since
XX*\A*BB*
A* . The
c
necessary condition is then proved.
Conversely, let
(P*,Q*)
be a partition of
be the basis of the associated monoids.
x*
We show that
conditions (i), (ii) and (iii) of proposition 30.
{e}
A* n B*
c
p* n Q*
c
= {e}.
This gives (i).
from lemma 33 and its sYmmetric version.
by induction on the length.
that it belongs to
B*.
any left factor w'
Q*.
satisfies
First
XX*\AX*.
We have to show
implies w' i AX*
for
Thus we conclude from lemma 33 that wand
is a partition of X*,
tors belong to
(A*,B*)
(X*\P*)*.
Since
XX*\p*
c
Q* because
we see that wand any of its left fac-
Hence w€B*.
Q.E.D.
In the beginning of this section we have introduced the monoids
A*,B* used in the Sparre Andersen equivalence principle.
that A*
(resp.
Crespo
Q*).
B*)
is indeed the right Crespo left) associate of p*
for-any nonempty right factor w" of
p*
P*,Q*,
We have observed
For instance A* is the set of all words w with
tional property:
B
Condition (ii) follows
The condition w i AX*
any of its left factors belong to
(P*,Q*)
let A and
To verify (iii) we then proceed
Let w belong to
of w.
and
(or Q*)
w as soon as w is in
p*
The pair
(P*,Q*)
(w") > 0
has an addi-
contains all the rearrangements of a word
(or Q*) .
perty is called an abeZian set.
W.
S
A subset of X* having this pro-
The previous theorem relating partitions
and factorizations holds however even when
p*
and
Q*
are not abelian
sets.
35. Definition. A partition (P*,Q*) of x* satisfies
equivaZenae ppinaipZe if there exists a bijection
p
th~
Sparpe Andepsen
of x* onto itself
-92-
with the following two properties
(i)
(ii)
to
pp*
p maps any rearrangement class onto itself;
for any word w the number IT(w) of left factors of w belonging
is equal to the length
Lp(w)
of the longest left factor of pew)
belonging to the rigth associate of P*.
Now the following question arises:
characterize the partitions
which satisfy the Sparre Anderson principle.
brought to this question.
pair
(P*,Q*)
(P*,Q*)
No definitive answer has been
However when p* and Q* are abelian sets, the
does satisfy definition 35, as shown in theorem 38 below.
comes from the fact that Ian Richa.rds' map remains bijective when
This
(P*,Q*)
an. abelian partition, but not necessarily otherWise.
From now on assume that
P*,Q*
are abelian sets.
As in (65) we can
define the map p by induction on the length as follows
pee)
pew)
(70)
= e and
= xp(w')
for w = w'x , xe:X
or p(w')x according as
weP* or we:Q* •
36.
Lemma.
The map
p
is a bijeation and sends any pearTangement aZass
C onto itself.
w
Proof.
Clearly
The fact that
pew)
p* and Q* are abelian sets plays a crucial role.
is a rearrangement of w.
to the same monoid
p* or Q*.
Thus wand pew)
Assume that
always belong
p is bijective when it is
restricted to the set of words shorter than w.
If pew) = veP*,
we
is
-93-
know that w€P*
with v
we have
and there exists one and only one pair
(x,w') € X* x X
= xv' , v' = pew') and w = w'x. In similar fashion,
w = w'x with p(w')x = v in a unique manner.
Q.E.D.
As was already noted in the previous section Ian Richards' map
introduced in (70) can also be defined as follows.
= V€Q*
if pew)
p
Let w = xl Xz
xn
iI' i Z' :~., i p (resp. iI' i Z' ••. i q ) the increasing
sequence of the subscripts i with the property that 1 s i s nand
(n > 0)
be a word,
XI X2 ' " Xi'"~P*
(resp.
(71)
pew) =
37.
Q*).
Proposition. Let A*
01 p*
(resp.
(resp.
Then
Q*).
Then
B*)
be the right (resp. ZeIt) assoaiate
pew) = ab with
(72)
Proof. The result is true if w has length 1.
on the length of w.
x in X.
Let
If wX€P*,
= ab, where a,b
Also
p(wx) = xab
pew)
then
an abelian set.
We cannot have xa € Q*,
would imply xab
€
P*.
then belongstoP*.
Hence xa € P*.
This shows that xa
with xa € A* and b€B*.
Again bx
is in Q*,
left factor of bx
Q*,
we have
p(wx)
Let us proceed by induction
given by (7Z), and
xab € p*
because with
€
A*.
is
this
We have then p(wx)
we have p(wx)
because otherwise we would have
= a(bx)
b~B*cQ*
p*
Any of the right factors", of xa
Finally, if wx € Q*,
is then in Q*
since
and so bx
€
B*
= abx
a(bx) € P*.
Hence, when wx
with a€A* and bx € B* •
= (xa)b
.
Any
is in
Q.E.D.
-94-
33. Theorem. When p* and Q* al"e abeZian
pX'incipZe hoZds foX' the paiX'
Proof.
sets~
the Spal"X'e Andersen
(P*,Q*) •
The theorem is a trivial consequence of proposition 37.
notations of (71) and (72) we have IT(w)
= p = lex . ...
1p
x.
1
With the
x.)
2
1
1
= lea) = Lp(w)
Q.E.D.
The content of this section is taken from Foata-SchUtzenberger (1971).
We could as well have deduced the results of this section from the very profound work by Viennot (1974) on·factoritations and constrUctions of free. Lie
algebras.
But then we would have been led to discuss various aspects of
Viennot's work and been very far from the topic of the present chapter
devoted to the constructions of rearrangement algorithms.
As a conclusion let us mention without proof the following theorem (see
Foata-Schtttzenberger (1971»
39. Theorem. Let
X
on the cpnstruction of partitions of x* .
be a finite set with k eZements
necessary and sufficient condition that
(P*,Q*)
(k
>
0) . A
be an abeZian pal"tition of
x* is that there exists a homomoX'phism 1.1 of x* into the additive gX'oup
of Rk and a Zexicographic oX'deX' s on Rk such that p*
and QQ*
= {w€X*:
1.1(w)
<
O} •
= {we:X*:
1.1(w) ~ O}
-95-
9.
Bohnenblust Rearrangement Theorem.
It seems that the only place where Bohnenblust rearrangement theorem
can be found is in a paper by Farrell (1965):
stated in terms of permutations.
The theorem is there
However there is a natural extension
to the multipermutation case, namely to the case of any rearrangement
class
C(w),
where the word
w may have repeated letters.
The pur-
pose of this section is to propose and discuss that extension.
Again, we start with a totally ordered set
monoid
X and form the free
x* generated by X. We recall that two words w,
are conjugate if w = uv
and
= vu
WI
WI
of x*
both hold for some words
u,v.
Clearly, conjugacy is an equivalence relation. The set of all words conjugate with a word w is called the conjugate cZass of w. A word
n
w is primitive if we cannot have w = v for some word v and n > 1 •
j
Let
P be the set of all primitive words of
qu~tient
set of
X*.
P derived by the conjugacy relation.
a bijection y of
C onto some subset S of
C for any C in
C
C the
The elements of
A cross-section of
C are then the primitive copjugate classes.
belongs to
We denote by
P such that
C is
y(C)
For instance, we can take as
y(C) ,
the minimum -with respect to the lexicographic order of X* - of the
elements in C.
We shall go back to this example later in this section.
Next, we form the free abeZian monoid
elements of
belong to
C+
C.
are monomiaZs
Two products
C1 Cz
C1 Cz
for some permutation
of 1 2 •• k .
a
generated by C.
where
Ci Ci
The
CI ' CZ' .•• , Ck
Ckl
represent
Ci C2 ... Ck = Ca (l) Ca (2) ••• CaCk)
=k
and
Ck '
Ck and
the same monomial, if k
l
C+
-96-
Let m ~ 2 and consider a sequence
partitions of X*.
P~
QJ~
and
J
This means that for any j
P~
J
= x*
u Q!
J
(Pi, Qi) , .•. ,
= 1,
(P~,~)
of m
2, ... , m the sets
x* satisfying the properties
are submonoids of
The sequence
(Pi, Qi) , .•. ,
P~ n Q~
J
(P~,~)
J
= {e}
•
is said to be an m-partition if
the further properties hold
P~ n P~
(i)
= U
J
Q~
J
,
i;tj
P~
for any
1
the complement of
P.P~
J J
for
= x*
j
1 s i < j S m
.
= 1,
= P~\{e},
J
2, ••. , m.
The assumption that
is a submonoid is essential in
For m = 2 we get back the notion of partition of X*.
the sequel.
An
= {e}
u ••• u P~
Pi
(ii)
Of course Q*.
J
1
(73)
example of an m-partition is obtained by taking a finite totally
x = {Xl' x2 ' ..• , xm} with m elements, and letting
be the set of all words in x* whose maximum letter is equal to
ordered set
X
j
P.P~
J J
(j = 1, 2, ... , rn) •
For any word w = Xl x2 .•. xn with positive length n and any subscript i (lsisn) we let y.1 = j if and only if the left factor
Xi
U(w)
(74)
and
of w,
Yew)
U(w)
with length i ,
belongs to
be the two two-row matrices
. [::
::
'"
::]
Yew)
Pj.
Then, we let
-97-
Theorem
40.
m-partition of x*
sets.
(Pi, Qi) , ... , (P;,
Let
(Bohnenblust).
and assume that
Then, there exists a bijection
properties:
section y
if
sew)
q;)
be an
Pi, ••. , p*m are aU ·ab.eZian·
13 :
is the monomial-
x*
+
C'"
having the foZZowing
Cl C2 .•. C ,
k
then for any cross-
of C
the word y(C l ) Y(C Z) ... y(Ck) is a rearrangement of w
the juxtaposition product Vy(C l ) . VY(C Z) . . . . . Vy(C k ) is a
rearrangement of the coZumns of U(w) .
(i)
(ii)
For example, consider the Sparre Andersen partition with
Then
is the set containing the empty word,
p* (=Q*)
1 Z
and the non-empty words
(resp.
~
the word
0).
X =R .
xn su'ch that xl'" X z ... ... +.xii > 0
Also consider the class C(w) of the lZ rearrangements of
xl
X
z
w = -2, -1, 1, 1.
In the following table those lZ rearrangements
occur as top rows of the matrices
U(w).
There are also lZ distinct
monomials
C C2 ..• Ck such that for any cross-section y of C the
l
word r(C ) Y(C )
y(Ck) belongs to C(w). In the following table
Z
l
with each word w is associated a juxtaposition product V(w l ) V(w Z)
which is a rearrangement of the columns of
is then a rearrangement of w
(i
= 1,Z, ••. ,k).
Then
Sew)
Let
= C1
C2
C.
1
U(w).
The product
wI
W
z
be the conjugate class of w.
1
Ck
V(wk )
wk
-98-
U(w)
[-2 -1 11]
[-2 -1 11]
[-2 1-1 1]
[-2 1-1 1]
[-2 11-1]
[-2 11-1]
[-1 -2 11]
[-~] [-~ ~ ~]
[-1 1-2 1]
[-~ ~) [-~ ~)
[-1 11-2]
[~) [-~ -~ ~)
-2 -1 1)
-2 1-1]
[:) [-~ ~) [-~]
[:) [-~ ~ .~)
-1 -2 1]
[:)[-~)[-~ ~)
2
222
2 2
2 2
222
2
2 '2 2 2
2 2
2 2
221
[:
[:
[:
r:
2
222
222
222
-1 1-2]
212
1
-2 -1)
[:
1
r:
1 -1
1
2
1
2
-~]
222 2
2 2
2 2
222
2
[:)[:)[-~)[-~]
[:) [:)[-~ -~]
[-1 11]
1 1;1
n
2
-99-
In order to prove theorem 40 it will be convenient to use the crosssection y: C + S,
where S
is the set of all standard lexicographic
words, introduced by Chen, Fox and
L~ldon
(1958).
As
X is supposed to
be totally ordered, the lexicographic order induced on
of
X makes of x*
x* by the order
A word w in x*
a totally ordered set.
is said
to be standard if it is (strictly) smaller (with respect to the lexicographic order in
In other words
u and v
~
e,
X*)
than any of its non-trivial cyclic rearrangements.
w is standard if for any factorization uv
we have w
<
vu.
This definition implies in particular
that a standard word is always primitive.
ard words.
Let S be the set of all stand-
For any primitive conjugate class
unique element of the set
C n S.
of w with
C we let
y(C)
be the
The proof of the next lemma can be
found in Chen, Fox and Lyndon (1958) and is not reproduced •
41.
Lemma. (Chen, Fox and Lyndon (1958)). The three foZZowing properties
are equivaZent
(i)
w is a standard word;
(ii)
either· w beZongs to
X,
or w = w'w" with w' , w19 both
standard and w' < wI! ;
(iii)
w < wI! for any non-empty proper right factor
wI!
of w •
From lemma 41 we can deduce the following proposition.
42.
Proposition. Any non-empty word admits a unique factorization
WI w ••• wp '
2
caUed its .lexicographic standard factorization having
-100-
the 10 ZZowing two properties
••• , w aPe standaX'd;
p
(i)
For example, the lexicographic standard factorization of
w = 5823524211312112 is
(58, 23524, 2, 11312, 112) •
Proof. A word with length one is always standard. Hence, any non-empty
word w has a left factor which is standard.
left factor of w with maximum length.
word wi.
Let
wI
be the standard
We have w = wI wi
for some
Again, we can consider the standard left factor
with maximum length.
w2 of wi
By continuing this procedure we obtain a sequence
•.• , wp of standard words whose juxtaposition product is
w.
If
1 s i < P and
wi < wi +1 ·, lemma 41 (ii) would imply that the
word w.1. w.1.+ 1 be standard. This would contradict the defnnition of w.•
1.
Hence (ii) also holds.
Conversely, suppose that there is another factorization
of w into standard words with the property that
Let
i
be the smallest integer with w.
length lCvi)
1
~
V.
1
VI
~
v
2
~
VI v 2 ••. vr
~
v
r
We may assume that the
of Vi
is less than l(w i ) . Thus Wi = viu. If
u = v.1.+ I ... V.l+q for some q ~ 1 we have Wi = Vi v i +l .•• v i +q •
Hence v.1. < w.1. < V.1.+q since w. is standard. This contradicts the
1.
V.
VI with q ~ I and
If U = v +
assumption v. ~ v.l+q
1.
l+q-l
i 1
VI a proper left factor of v.
, we have v i +q = vlv" with v I ,v" ;t e
l+q
Then v.1 < w. < v' < v.1.+q since w.1 °is standard and v.1 and VI are
1.
proper left factors of w.1. and v i +q , respectively. We have again
.
...
.
-101a contradiction.
Hence there can be only one non-increasing factoriza-
Q.E.D.
tion of w into standard words.
With each pair
(Pj, Qj)
we associate the factorization
each j
= 1 , ••• ,
ments of
J
determined by theorem 34.
(A~, B~)
J
J
m we denote by A!
J
(resp.
A~
(Pi, Qi) , .•• , (P;,
of the m-partition
(resp.
~)
For
the set of all ele-
B!)
J
which are less (with respect to the 1exico-
B~)
J
graphic order) than any of their non-trivial cyclic rearrangements belonging to
(resp.
A~
J
are primitive.
43.
J
J
(pesp.
jugate aZass of w aPe
If a a 2 ••• am
l
is the element of A' (resp.
A*.(r~sp.-B*)
B*)
Q*)
but no conjugate in
B*
B')
in the Gon-
= 1,2, .•• ,m-l)
a.~+ 1 •.• am a l ••. a.~ (i
Proof. Proposition 32 implies that a word w in p*
(resp.
(pesp.
B').
0, aI' a 2, ..• , am € A (resp. B))
aonjugate lUith w, then- the·'otner eZements of
>
conjugate in A*
J
A~
The aonjugate aZass of a primitive lUopd w of p*
Lemma.
B!)
In lemma 43 and proposition 44 below we write A, A:. A* ...
aontains exaatZy one element of A'
(m
(resp.
J
J
instead of A., A!,
J
In particular, the elements of A!
B~).
(resp.
(resp.
.
Q*)
has a
A*).
This proves the first part of the lemma, since the element in A'
(resp.
is the minimum among all those elements in the conjugate class of w,
which belong to A*
Let
A'
a
B*).
a l a 2 ••• am (m > 0, aI' a 2, ••• , am € A) be the element of
in the conjugate class of w (the "resp." part is omitted in the
=
rest of the proof).
w'
(resp.
= va.
~+
Any other conjugate of w has the form
1 ••. am a l •.• a i - IU with wv
=
a 1.•
If w'
belongs to A*,
B')
-102-
we have u e A*
v
=e
.
since A*
Thus w'
the latter form
44.
is right prefix.
As uveA
J
we must have
= a'+
1 l
(1
~
... am a 1 ••. a.1- 1 a.
. Conversely, any word of
1
i ~ m - 1) belongs to A* .
Q.E.D.
Proposition. Any 1iJor'd w in A* faator'izes in a unique manner' as
a non-inar'easing pr'oduat of eZements of A' •
Proof.
by A.
From 1enuna 29 it follows that the monoid
y*
t'
is freely generated
With the lexicographic order induced by the total order of· X the set
A becomes totally ordered.
y,
A*
in Y llet
y
<
generated by Y.
6: Y + A be any bijection, and for
if 6(y)
y'
<
6(y')
Then form the free monoid
For any elements Y1' Y2' •.• , Ys (s > 0)
Ys + 6 (Y1) 6(Y2) ..• 6(ys)
the map 6: Y1 Y2
y* onto A*.
Let
is an isomorphism of
We can consider the set T of standard words in y* •
According to proposition 42 any word of y*
can be expressed uniquely
as a non-increasing product of standard words in Y*.
In its turn, the set Ali
by
a.
= 6(T)
•
factorizes in a unique manner as a
non-increasing product of elements of A".
if we show that the sets
A'
inverse of 6,
and let
a
element of A".
Then
= 1,
Let A"
is totally ordered by the order of T induced
Hence, any word w in A*
for any i
in Y
... , m·· -; 1 .•
The proof will be completed
and A" are identical.
= al
Denote by
a 2 •.. am (aI' a 2, .•• , am e A)
We cannot have
al
= a 1·+1,
a2
n the
be an
= a.1+2"'"
a·m-1.
= a1;
-103-
for any i
and so
that
with 1
~
i
~
m - 1,
because (75) would imply
n(am_i +1 ) ••• n(am) < neal) ••• n(am), contradicting the fact
neal) ••• n(am) is standard. Hence for any i with 1 ~ i ~ m - 1
there is an integer j
m - i , a i +k = a k
a. • Again (75) impltes that n(a .) < n(a. . )
with the property that
for any k < j
1
~
j
~
and a i +j ~
J
J
and so a. < a.1+ j
Now a j cannot be a left factor of a i +j ,
J
both words belong to A and A* is right prefix. Hence
1+)
.
a. . .. a
J
m
<
because
...
a. . ...
1+)
Consequently, the word a
a belongs to A'. Thus A"
m
l
The reverse inclusion is proved as follows. Let a a
l 2
c
a. .
1
A' •
...
a
and
m
., ai, • • • I a' -€ A)
, am, aI'
a'1 a'2
a'n be two words of A* (aI' a 2 ,
n
with a a
a < a' a'
As no element of A can be a left
a'n
1 2
m
1 2
factor of another element of A , there is an integer j with the pro-
...
...
...
perty that
.
1 ~ j ~ min(m,n) , a
Moreover, we can write
letter of
v
i
a = uv,aj
j
= ai for any
= uv' with
and a j ~ aj •
v,v' non empty, the first
i
< j
being different from the first letter of v' .
Then
a'n
The first letter of v is then less than that of v'
and a.1 = uv < uv'
= a'j ; .
Consequently A'
A" •
c
As
a.1
= a!1.
for
i < j
Thus v < v'
this shows that
Q.E.D.
-104-
We are now ready to complete the proof of theorem 40.
are abelian sets, we can define the Ian Richards' map
to the pair
(Pj,
Qj)'
be a non-empty word of
P~
J
x· .
i(j,l), i(j,2), ••• , i(j,p.)
J
i
with the property that
i(j,h)-th
U(w)
column
The symbols
as we did in (70).
simply to be replaced by
and
respectively.
Q~
J
For any
= 1,
j
Pj
As
and
P~
J
Q~
J
corresponding
p*
and
Let w
= Xl
Q*
are
x2
••• , m we denote by
the increasing sequence of the subscripts
1 ~ i ~ n
and Xl x 2 •.• Xi € Pj. The
(h = 1, 2, ••. , Pj ; j = 1, 2, •.• , m) of the matrix
is then
Let
(76)
Proposition 37 says that
belongs to
A~.
J
aj
is the longest left factor of Pj(w)
In particular, as
a.
J
€
that
we have
P~,
J
...
so that the juxtaposition product Veal) v(a z)
ment of the columns of the matrix
... ,
(78)
with
a1
€
Ai
b2 with
a2
€
Ai
bl
V(am)
a
is the following
is a rearrange-
U(w) •
An alternate way of defining
= a1
= a2
...
m
-105-
This can be proved as follows.
Suppose xl ••• x.
1
with
€ P~
J
j > 1 •
= P1(x1· •. xi_I)xi' Let P1(x I •.. xi _l ) = ab with
since
a € Pi ' b € Qi. The word P1 (x I... xi ) = abxi belongs to Pj
x.1 . As Pi c Qj' the word a is in
it is a rearrangement of Xl
Q~. But the relations a € Q~ , abx.
P~ imply that bX € Pj . Now
i
J
J
1
J
PI (x 1 ···x i )
Then
€
= a'abxib ' with a l = ala Pi and
p.(b l ) = a. b! with a. given by (76) and
J
J J
J
P1(w)
= bXib'
bl
€
b!
J
€
Qi.
for any
€ Q~
J
Hence
j > 1 •
Having obtained the sequence
(aI' a 2, •.• , am) either by (76) or
as a non-increasing product a(j,l) ••. a(j,r j )
(78) we factorize each a j
of elements of Aj (j = 1,2, ••. ,m)
according to proposition 44.
of the words
belongs to
a(j,l), ••• , a(j,r j )
V(a.)
J
= V(a(j,I))
Pj,
As each
we have
••• V(a(j,r.)) •
J
The juxtaposition product
V(a(m,l)) ••. V(a(m,rm))
V(a(I,l)) •.• V(a(l,r l ))
is then a rearrangement of the columns of U(w)
conjugate class of a(j,s) (s
= 1,
sew)
(79)
•.. , r j ; j
= ITC(j,s)
For any word v
c~njugate
an abelian set.
In particular V(v)
with
a(j,s)
columns of the matrix V(a(j,s)).
the juxtaposition product
U(~'l)
•
nVyC(j,s)
= 1,
Let
C(j,s)
... , m)
be the
. We then put
•
we have v
€ P~
J
since
P~
J
is
is a (cyclic) rearrangement of the
Hence for any cross-section y of C
is a rearrangement of the columns of
-106-
45.
Example.
Let
w be the word
For any integer
j
words
• •• x
is
j
in the interval
m = 7).
-3, 2, 3, -1, 3, 0, -3, 4, -2, 4, 2 •
consider the set
[-3,4 ]
(P~,
Q~)
J
J
(j
= -3,
••. , 4)
of the
is an m-partition (with
-3 2 3 -1 3
= [:
123
° -3 4 -2 4 2]
333
3 4
444
With the same notations as (76) we see that the words
are empty.
a_ , a_ Z' a_I' a
3
O
Furthermore
veal)
V(a4 )
a3
J
Here
U(w)
Now
P~
such that the maximum of the letters
n
The sequence
.
= 1,
w
= -3,
= [i)
V(a Z)
= [2
4-2 4]
4 4
= [;]
is greater than
[-: ° :]
3 -1
3 3
3
4 4
0, 3, -1, 3 belongs to
rearrangement belonging to
V(a3 ) =
A~
is
3,
A
because the other cyclic
-1, 3, -3, 0, 3,
and the latter word
a
• Finally the factorization of a
as a non-increasing
3
4
With ~(v) denoting the
product1 of elements of A is (2,4), (-2,4)
4
conjugate class of the word v,
sew)
= f(l)
we have
f(2) r(-3,0,3,-1,3) f(2,4) r(-2,4) .
Conversely, let
C C ••• C be a commutative juxtaposition product
I 2
k
of primitive conjugate classes. As the submonoids Pj (j = 1, .•• , m) are
abelian sets, each class
one
P~.
J
C.1 (i
= 1,
••• , k)
It follows from lemma 43 that
C.
1
is contained in one and only
contains one and only one
-107-
element of Aj,
all the words
say wi • We next form the non-increasing product of
wi which belong to Aj .
Let
a.
J
be this product.
The
word w corresponding to Cl C ••• C is obtained by the equations
2
k
(80)
We can also deduce from theorem 40 the following result on sequence
rearrangements.
of X*.
Again suppose we have an m-partition
For any w in X* we define
U(w)
(Pi, Qi)' ••• ,
and V(w)
(P~, ~)
as in theorem 40.
(wI' w2' ... wp ) be the standard lexicographic factorization of w .
Then VIew) is defined as the j\~taposition product
Let
(81)
45. Theorem. Thepe exists a bijection a' of x* onto itseZf
foZZo~ing
prope:t'ty:
~ith
the
fo:t' any w in x* the juztaposition product VI (fl(w))
is a :t'eaP:t'angement of the coZumns of U(w) •
Proof. Consider the bijection 8: w + Cl C2 ••• Ck of theorem 40. Each
primitive class
Ci contains one and only one standard word, say wi' As
the way of writing theCl's is immaterial, we can suppose that they are
labeled in such a way that WI
~
w2
~
...
~
wk'
It follows from proposi-
-108-
tion 42 that there is one and only one word
graphic factorization is
5' (w) =
W'
(WI' w2 , ••• , wk ) • Then we only have to put
Q.E.D.
•
Let us take up again example 45.
4, -2, 4, 2 and
B(w)
= f(l)
We had w = 1, -3, 2, 3, -1, 3, 0, -3,
f(Z) f(-3,0,3,-1,3) f(2,4) f(-2,4).
standard words occurring in each class are
and
whose standard lexico-
W'
(-2,4).
The
(1), (2), (-3,0,3,-1,3), (2,4)
When rearranging them in non-increasing order we obtain
(2,4) (Z), (1), (-2,4), (-3,0,3,-1,3) .
Thus
B' (w)
46.
~emark.
= Z,
4, 2, 1, -2, 4, -3, 0, 3, -1, 3 .
SchUtzenberger (1965) has introduced and used the following
notion of factorizations of free monoids:
and
(T,o)
let T be a subset of words
a an injection of T into a totally ordered set J.
is said to be an S-factopization if any word w in
que factorization
(i)
(82)
(it)
The pair
x* has a uni-
(WI' w2 ' ••• , wp ) with the following properties:
WI' wz,
... ,
wp belong to T
o(w );:: a(w ) ;:: ...
l
2
~
0(,\»
•
Of course, the standard lexicographic factorization is an S-factorization.
As was shown by SchUtzenberger (1965), any S-factorization
(T,a)
has the
property that any primitive conjugate class contains one and only one element of T.
Hence theorem 45 holds for any S-factorization
(r,o).
We only
in (81) by assuming that WI' w2 ' ••• , wp
are words of T satisfying condition (ii) of (82).
have to change the definition of V'
-109-
APPENDIX
2 and w = xl x2 ••• xn be a permutation of 1 2 ••. n •
The up-down sequence (or variation, or still pattern) of w is a sequence
Let n
....
~
Yl Y2 ••• Yn - l of plus or minus signs defined by Yi = + (resp. -) if
Xi < (resp. » xi +l ' Niven (1968), Car1itz (1973), Foulkes (1975) have
brought complete answers to the following question: how many permutations
are there admitting a given sequence v of plus or minus signs as their
up-down sequences? For instance, the permutations whose up-down sequences
are
- + - + •••
are called alternating.
Since D6sir6 Andr6 (1881) it
is known that the number of alternating permutations of 1 2 ••• n is
equal to the coefficient
Dn
of un Inl
sec u + tan u
=1
+
in the expansion of
r (un/n!)Dn
n>O
•
Dn is called the tar~ent (resp. seaant or Eule~)
It is easy to concelve that the alternating case,
For n odd (resp. even)
number of order n.
together with the two trivial cases
- - - •••
"extremal points" of this counting problem.
given formulas for the general case.
and
+ + + •••
are the
The three above authors have
Niven's formula has a determinental
form; that of Carlitz is given in terms of an alternate-sum that: gets ,<very
simple for special patterns; finally, Foulkes' formula involves the coefficients
gW\.ll\~
occurring in Schur function multiplications. Although there is
a well-known rule to calculate gW\.lA
(namely, the Littlewood-Richardson
rule), the algorithm is not easy to handle and Foulkes' formula may seem
to be of a limited U$e.
is found between
However, it is the first time that a connection
syw~etric
group representation theory and traditional
-110-
methods of permutation enumerating.
a new theory on counting.
This connection might give raise to
In the rest of the present appendix, we would
like to state Foulkes' result only.
Let v
= vI Vz ...
vn _l be a sequence of plus or minus signs. It
corresponds to v a partition (A) of some integer (m+n) in the fo1lowing manner:
put
of integers
i
o = + and let iI' ••. , i r be the increasing sequence
V
such that
o sis
n - I
and v.~
= +.
Then
(A)
is the
partition
(AI' A2 , ••• , Ar ) with Aj equal to one plus the number of
minus signs in Vo vI
vn-l on the right of v.~.
For instance,
with v
and
=+ + -
(AI, ••• ,A s)
J
we have r = 5 , (il, •.• ,i s ) = (0,1,2,5,7)
= (5,5,5,3,2) = (5 3 ,3,2). Such a partition is repre-
- + - + -
sented by a Young diagram having
nodes on the r-th row.
In a Young
diagr~~
Al
nodes on the first row
For instance, the Young diagram of
, ••• , Ar
(5 3 ,3,2) is
the nodes placed on the extreme right of a row, or at
the bottom of a column form the 'lim of the diagram.
diagram associated to a
se~lence
exactly n nodes on its rim.
v of
(n-l)
Clearly, the Young
plus or minus signs has
Furthermore, when running along the rim
from right to left and top to bottrnn, starting with the right-most node
of the first column, we recover the sequence v by associating a minus
to a "flat" and a plus to a "down".
VI'hen removing the rim out of the
-111-
Young diagram of
(A)
we obtain a second diagram which is the Young dia-
(00)
gram of a new partition l say
removed exactly n nodes.
I
of the integer m I
In the above example
can now characterize any sequence v ov
an ordered pair
((A)I (00))
(n-l)
as we have
We
(00)
plus or minus signs by
of partitions; the Young diagram of
has exactly n nodes on its rim and
diagram is obtained from that of
(A)
is the partition whose Young
(w)
by removing the n nodes of the
(A)
rim.
Theorem (Foulkes (1975)). Let ((A)I (00)) be the ordered pair of partitions corresponding to a sequence v of (n-l)
pZus or minus signs.
Then,
the nwnber of pe:t'1TiUtations with up-down sequence v is equaZ to
r gOOll A f(A)
I
II
where the summation is extended over aZZ partitions
above formuZa fell)
(ll)
of n.
In the
is the dimension of the irreducibZe representation of
the symmetric group Sn corresponding to
cient of the Schur function
{A}
(ll)
and goollA is the coeffi-
in the produat {oo} {ll} •
Until Foulkes's paper is pub1ished l it is perhaps fair to give a few
references.
First l there is an excellent survey paper by Foulkes (1974) on
"classic" algebra.
The most elementary expose' on Schur functions can be
found in Littlewood (1970) (chapter 6).
product of two Schur functions
{oo}
combination of Schur functions l i.e.
and
See also Stanley (1971).
{ll}
The
can be written as a linear
-112-
What is most remarkable is the fact that the coefficients g ,
l.dllA
negative integers.
Littlewood-Richardson rule that describes the compu-
tation of the integer
(chapter 15).
Thrall (1954».
are non
g ,
l.dllA
is clearly explained in Littlewood (1970)
There is an explicit formula for
The coefficient·
Young tableaux with shape
ell)
fell)
f(u)
(fee Frame, Robinson,
also counts the number of standard
(see Littlewood (l940) p. 68), or Stanley
(1971)) •
In his coming paper Foulkes also proposes interesting extensions of
Call inversion sequence of a permutation w the updown sequence of the inverse w- l • Then Foulkes gives a formula for the
the above theorem.
number of permutations whose up-down and 'inversion sequences are prescribed.
-113-
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"""'"
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"""
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