PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 45, Number 2, August 1974
ENUMERATIONOF POSETS GENERATED BY
DISJOINT UNIONSAND ORDINAL SUMSi
RICHARD P. STANLEY
ABSTRACT.
be
built
of disjoint
tained
Let
union
for the
asymptotic
be the number
collection
and ordinal
generating
estimate
lem for labeled
Let
f
'n
up from a given
posets
P and
sum.
relations
union
P + Q is defined
using
functional
Using
be given
which
can
the operations
equation
a result
is ob-
of Bender,
an
analogous
prob-
Regard
P and
for f . The
considered.
Q be partially
being
posets
posets
Sfx.
sometimes
is also
n-element
A curious
function
can
of
of finite
ordered
on two disjoint
sets
sets
(or posets).
T and
to be the partial
T , respectively.
ordering
The
on TuT
Q as
disjoint
satisfying:
(1)
If x € T, y eT, and x < y in P, then x < y in P + Q; (2) if x € T', y € T', and
x<y
in Q then
the partial
tion
x <y
ordering
in P + Q. The ordinal
on T U T
(3) if x e T and y eT
satisfying
, then
sum
P©Q
is defined
to be
(1), (2), and the additional
x < y in P©Q.
Hence
condi-
+ is commutative
but © is not.
The question
isomorphically
joint union
we consider
distinct
or an ordinal
(+, © ^-irreducible.)
« can be built
and ordinal
Hence
the
such
if P and
Call
a poset
Q are
(+, © )-irreducible
if S consists
5-posets,
of a single
n denotes
P of « copies
-
S be a set of nonvoid
no element
posets.
isomorphically
P of S is a dis-
(We say that
distinct
posets
of S by the operations
that
5-posets
Let
that
sum of two nonvoid
up from the elements
sum?
3. Here
posets,
How many
can be obtained
then so are
are simply
one-element
5-posets and five 3-element
and
is the following.
finite
poset,
5-posets,
an «-element
members
then
of disjoint
in this
P + Q and
the
there
and
«P
union
way an S-poset.
PffiQ.
Moreover,
of S. For instance,
are two 2-element
viz., 21, 2, 31, 2+1,
chain
P is
of cardinality
(21) © 1, 1 ©(21),
a disjoint
union
P +•••
+
of P.
Received
by the editors April 23, 1973.
AMS(MOS) subject classifications
(1970).
Key words
generating
'The
and phrases.
function,
research
of California
Polya's
was
Poset,
partially
enumeration
supported
Primary 05A15; Secondary 06A10.
ordered
theorem,
by a Miller
set,
functional
Research
disjoint
union,
ordinal
sum,
equation.
Fellowship
at the University
at Berkeley.
Copyright © 1974, American Mathematical Society
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295
296
R. P. STANLEY
Let
Define
/
denote
the number
the generating
tional
equation
for F(x).
in [3, Chapter
ever,
arising
P . is void,
and
essentially
we agree
every
that
«-element
the
a func-
to that
appearing
networks.
us to obtain-an
explicit
by the triviality
Howfunc-
of one of the
of «-element
fine the generating
v
where
By convention
© .
of S,' so an0 = 0 since
Let
u
be the number
of «-element
/ = u n + v n -a
>n
is
nei-
+ and essentially
the number
un00 = v„ = 0. Hence
P.
not a member
© , but not both.
members
and
5-poset
essentially
of S to be nonvoid.
+ 5-posets
We define
neither
if P = Pj©P2
© . Every
of S is both
members
P l + P 2 where
+. Similarly
+ or essentially
member
essentially
© S-posets.
r
/_ = 1.
determine
is analogous
P can be written
P is essentially
a n be the number
are assuming
«. We set
We shall
of series-parallel
are helped
P is essentially
we say
of S is either
Let
used
allowing
we instead
x".
from the enumeration.
then we say
ther
principle
for F(x),
If P is an S-poset
void,
The technique
of a duality
equation
two groups
of cardinality
F{x) = 2°°_n/
6, § 10] for the enumeration
instead
tional
of 5-posets
function
we
of
essentially
if « >
1. Den
—
functions
oo
A(x) = £
anxn,
U{x) = £
72=0
It follows that
(1)
Now every
essentially
as a disjoint
5-posets
r
P.,2 where
P
P.'s
i
is immaterial.
of cardinality
Define
to 5 can be written
of m > 2 essentially
« which
Let
of addition
from Pólya's
be the num-
are the disjoint
for m > 2 the generating
in a disjoint
theorem,
union
as expounded
is immaterial,
in [3, Chapter
Um (x) = Z(<5m \V(x),
V(x2), V(x5), ., . ),
'
\z,,
u mn
©
where
Z(6
group
°
r
S 772 of degree
°
functions
z2, z,,
■• •) is the cycle
m. Now
index
polynomial
A(x) + 2°°m~¿ ,(7 772(x) = U(x).
it follows
6], that
m—
> 2,
of the symmetric
Hence
from (2) we
obtain
oo
(3)
U(x) = A(x) + £
Z(em| V(x), V{x2), V(x3), • • • ) - V(x) - 1,
772=0
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since
union
xn.
the order
(2)
v '
P not belonging
of the
© S-posets.
n= I mn
immediately
v%*.
22=0
P.+P2+...+
+ 5-posets
of m essentially
Since
+ 5-poset
union
the order
ber of essentially
m
V(x) = £
Fix) = U(x) + V(x) - A(x) + 1.
uniquely
U (x) = 2°° ,u
unxn,
22=0
Z(<5\V{x), V(x2),..-)=
1 and Z(6j| V(x), V(x2), ■■■) = V(x).
297
ENUMERATION OF POSETS
Now it is well known that
£
Z(6Jzj.
zr zy ...)tm
=expU1z'+z2r2/2+
z^/3
+ ■• • )
772=0
(cf. [3, p. 133]). Thus from (3) we get
V(xk)
z
U{x) = exp
(4)
V(x) + A(x) - 1.
k=l
Similarly
written
every
uniquely
+' S-posets
r
essentially
P.,2> where
be altered,
sum
of the
to u 72222 and
we get
P not belonging
P1©P2©
now the order
V 772(x) for ?« —
> 2 in analogy0/
ming cannot
© 5-poset
as an ordinal
• • ■ ©F
P.'s2
is fixed.
Í7772(x).
from Pólya's
Since
theorem
where
Z(G
group
G
\z.,
z 2, z,,...)
of order
is the cycle
one acting
index
Define
of sum-
that
m > 2,
polynomial
on an ?7z-set. Since
v 77272 and
now the order
V (x) = Z{G | U(x), U(x2), C/(x3), . . , ),
(5)
to S can be
of m > 2 essentially
Z(G
\z,,
of the trivial
z,,...)
= z". , we
get
(6)
Vjic) = l/to»,
Of course
(6) can be easily
the similarity
obtained
TO>2.
directly,
but we wanted
to make
of (2) and (5).
Now
OO
V(x) = A(x) + £
772=2
(7)
Eliminating
OO
Vjx)
= £
U{x)m - U(x) + A(x) - 1,
772=0
V(x) = 1/(1 - U(x)) - U(x) + A{x) - 1.
U(x) from (1) and (7) yields
(8)
V(x) = Fix) + l/F(x)
-2
+ A{x).
Thus by (4) and (8),
Fix) = Uix) + Vix) - A(x) + 1
exp
[ÈÂ^
1
Fixk)
- 2 + Aixk)')
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We have obtained
clear
298
R. P. STANLEY
Theorem
1. F(x)
(9)
satisfies
[bh
„k\
Fix) = exp
Since
exp (2°° ax
k= 1
form
curious
the functional
/k) = (1 - x)
.
equation
1
Atjk
- 2T +. Aix«)
)
Fixk)
a, (9) may be rewritten
in the rather
-(/,+g.+a,)
f(x)= nci-*f)
where l/F(x) = 1 + 2Tg
x1.
1
2
For instance,
if S consists
of a single
one-element
poset,
then
A(x) =
x and
F(x) = 1 + x + 2x2 + 5*3 + 15*4 + 48x5 + 167*6 + • ■•.
Another
finite
case
of interest
is when
(+, ©)-irreducible
distinct
posets
S consists
posets.
of cardinality
Then
«.
of all isomorphically
/
is the number
It can be shown
(cf.
distinct
of isomorphically
[4] for the numbers
/ ) that here
' n
A(x) = x + x4 + 12x5 + 104x6 + 956x7 + . • •,
Fix) = 1 + x + 2x2 + 5x3 + 16x4 + 63%5 + 318x6 + 2045*7 + •«'•.
A theorem
asymptotic
reader
of Bender
formula
the details
we get /
and
when
5] allows
a
the determination
A(%) is a polynomial.
of the calculations
~ Cn
(1 + \/5)/2,
[1, Theorem
for /
", where
and merely
state
a is the unique
C is a constant
given
of an
We shall
that
positive
spare
when
the
A(x) = x,
root of F(a)
-
by
."TV/2
1
i7r(3v/5-5)
Note that not surprisingly
posets
of cardinality
One can also
of a finite
poset
P ate equivalent
ing injection
equivalent
»-element
members
Z *kF
1
/
k=2
\
is very much smaller
than
the total
by p
= 2n '
n, which
ask
by [2] is given
analogous
questions
P is an injection
if there
o: Z -> Z such
that
inequivalent
labeled
to any
(+, © )-irreducible
member
of S. (We set
posets
finite
component
of 5 (so
bQ = 0).
A labeling
Define
(ft and
Let
h
ift, of
that
b
the exponential
in-
be the number
can be built
© , such
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ment members
Let
of P is equivalent
A„ = 1 by convention.)
of
°
posets.
S be a set of nonvoid
posets.
(P, cf>) that
of + and
p
p: P -> P and an order-preserv-
cj>— cnjjp. Let
labeled
of S by the operations
for labeled
number
</>:P -» Z. Two labelings,
is an automorphism
(+, © )-irreducible
F(ak)2J_
up from the
the restriction
of 4>
to the labeling
be the number
generating
of
of a
of «-ele-
functions
ENUMERATION OF POSETS
Bix)=Z
bnXl,
M*)=Z
72=0
Then
by an argument
omit,
one obtains
Theorem
299
hmir
22=0
analogous
to that
2, H(x) satisfies
used
to prove
the functional
Hix) = exp[Hix)
+ l/Hix)
Theorem
I, which
w<
equation
- 2 + B(x)].
REFERENCES
1. E. A. Bender,
2. D. Kleitman
Asymptotic
Math. Soc. 25(1970), 276-282.
3. J. Riordan,
4. J. Wright,
Cycle
University
indices
in enumeration,
The number
SIAM Rev.
of finite
topologies,
analysis,
Wiley,
(to appear).
Proc.
to combinatorial
New York;
MR 20 #3077.
of certain
of Rochester,
classes
Rochester,
of quasiorder
N. Y.,
types
or topologies,
1972.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY,
CALIFORNIA 94720
Current
Technology,
address:
Department
Cambridge,
Amer.
MR 40 #7157.
An introduction
Chapman & Hall, London, 1958.
Dissertation,
methods
and B. Rothschild,
of Mathematics,
Massachusetts
Massachusetts
02139
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Institute
of