Cent. Eur. J. Math. • 10(2) • 2012 • 788-796
DOI: 10.2478/s11533-011-0100-5
Central European Journal of Mathematics
Compositions of n as alternating sequences of weakly
increasing and strictly decreasing partitions
Research Article
Aubrey Blecher1∗ , Charlotte Brennan2† , Toufik Mansour3‡
1 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
2 The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics,
University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
3 Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
Received 31 May 2011; accepted 21 September 2011
Abstract: Compositions and partitions of positive integers are often studied in separate frameworks where partitions are
given by q-series generating functions and compositions exhibiting specific patterns are designated by generating
functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and
strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such
partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”.
From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned
pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally,
we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or
a strictly decreasing partition.
MSC:
05A15, 05A16, 60C05
Keywords: Compositions • Partitions • Generating functions
© Versita Sp. z o.o.
1.
Introduction
Compositions and partitions of integers were extensively studied by MacMahon [5, vol. 2]. A survey of partition theory
is found in [1] and a survey of composition theory in [4]. Partitions have not generally been studied in terms of patterns.
∗
†
‡
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
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As in [4], composition generating functions often have occurrences of particular patterns as one variable. So compositions
and partitions have usually been studied in separate theoretical frameworks. Recently some work has been done on
expressing composition generating functions in terms of partition generating functions. For example in [2] a generating
function for a certain class of compositions, “concave” compositions having one “valley” pattern (and no peaks), was
developed. In [7, Chapter 2, Section 2.5] a similar idea was done for another type of “unimodal” compositions having a
single “peak” pattern (and no valleys).
Here, we consider arbitrary compositions and develop a generating function in terms of the size of the composition,
the number of parts and the total number of peaks and valleys. In other words any composition may be viewed as
an alternating sequence of “increasing” or “decreasing” partition blocks. This depends on precisely how “increasing”
and “decreasing” are defined. The generating function which we obtain, accounts for how many partition blocks each
composition of n with m parts splits into.
In Section 2 of this paper, we specify increasing partition blocks as those which are non-decreasing, in other words
weakly increasing, and decreasing partition blocks as those which are strictly decreasing. With these specifications an
arbitrary composition is split into an alternating sequence of increasing/decreasing partitions precisely at the part of
the composition in which the midpoint of any of the following patterns occurs:
TYPE
abc PATTERN DESCRIPTION
Strict peaks
Weak peak
Strict valleys
Weak valley
121
c=a<b
231
c<a<b
132
a<c<b
221
c<a=b
212
b<a=c
213
b<a<c
312
b<c<a
211
c=b<a
P P
Our main result is the generating function F (x, y, q) = n≥0 nm=0 x n ym qr , where x marks the size of the composition,
y the number of parts and q the number of occurrences of any of the patterns stated above. From this we deduce the
generating function for the total number of peaks and valleys in an arbitrary composition of n with m parts.
As an illustration, we may represent an arbitrary composition as a zigzag graph where every left to right increasing block
represents a non-decreasing partition with a line going up and where every left to right decreasing block represents a
strictly decreasing partition with a line going down. The composition 234343212345 of 36, is illustrated below.
@
@
@
@
@
@
@
In this composition, there are 2 peaks, 2 valleys and 5 partition blocks. The joining point between the blocks is either
at a peak or a valley. Hence, the number of such partition blocks equals the number of peaks plus the number of valleys
plus one. We now turn to the precise definitions required to develop the generating functions.
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Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions
2.
Definitions
We need the following definitions.
A composition σ = σ1 σ2 . . . σm of a positive integer n is an ordered collection of one or more positive integers whose sum
is n. Each summand σi is called a part of the composition.
A partition of a positive integer n is either a non-increasing or a non-decreasing composition of n. For example, for the
partition of n with k parts
n = a1 + a2 + · · · + ak
either
a1 ≥ a2 ≥ . . . ≥ ak ≥ 1
1 ≤ a1 ≤ a2 ≤ . . . ≤ ak .
or
As per the definition found in [6], let [n] = {1, 2, . . . , n} and let [n]l denote the set of words of length l in the alphabet [n].
For any word σ in [n]m , let red σ denote the member of [n]m obtained by replacing the smallest letter of σ by 1, replacing
all letters corresponding to the second smallest element of σ by 2 and so on. For example: if σ = 42244 ∈ [4]5 then
red σ = 21122.
We call {red σ : σ ∈ [n]m , 1 ≤ m ≤ l} the set of subword patterns in [n]l . We also say that there is an occurrence of the
pattern τ = σ1 , σ2 , . . . , σm at index i in the composition or word α = α1 α2 . . . αs if red (α1 , αi+1 , . . . , αi+m−1 ) = τ, where
i ≤ s + 1 − m.
Consider arbitrary compositions of n. We split them into alternating blocks of weakly increasing (non-decreasing) or
strictly decreasing partitions each of maximum size. The first block is either increasing or decreasing according to which
is the larger of the two. Thereafter, we alternate the blocks.
For example, consider the composition of 25: 3 3 4212541. It can be split correctly as (334)(21)(25)(41) starting with a
weakly increasing partition and incorrectly as (3)(34)(21)(25)(41) starting with a strictly decreasing block. The former
is chosen as it has a larger initial block.
We define peaks and valleys as in Section 1.
We introduce the generating function
F (x, y, q) =
n
XX
x n ym qr ,
n≥0 m=0
where x marks the size of the composition, y the number of parts and q the number of occurrences of any of the above
patterns stated in Section 1. We also define Fa = Fa (x, y, q) to be the generating function for all compositions of n
where the first part is a. Hence, considering all possible starting value for a, we have
X
F (x, y, q) = 1 +
Fa
a≥1
which includes the empty composition. We extend this notation to Fa1 a2 ...as as the generating function for the compositions
starting with a1 a2 . . . as .
Consider the pair of letters aj. We have two cases depending on the value of j. Firstly, if j < a, we define La to be
the generating function for all compositions for which there is a strict descent following the initial letter a. Secondly, if
j ≥ a, we define Ma to be the generating function describing a weak ascent following a. Thus
X
La =
Faj
(1)
j<a
and
Ma =
X
Faj .
(2)
j≥a
Putting all the possible cases together, we have the generating function for all compositions starting with the letter a:
Fa = x a y + La + Ma ,
(3)
where the first term is for a one part composition.
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3.
Generating functions
In this section, we find the generating function Faj for all words beginning with aj where j < a and Faj for all words
beginning with aj where j ≥ a.
Let j < a,
Faj = x a+j y2 +
X
Faji +
i<j
and similarly let j ≥ a,
X
Faji = x a+j y2 + x a y (Lj + qMj ),
(4)
i≥j
Faj = x a+j y2 + x a y (qLj + Mj ).
(5)
We substitute (4) into (1) and (5) into (2) to obtain
La =
a−1
X
x a+j y2 + x a y
j=1
and
Ma =
∞
X
a−1
X
(Lj + qMj )
(6)
(qLj + Mj ).
(7)
j=1
x a+j y2 + x a y
j=a
∞
X
j=a
We now define the following generating functions:
F (t) = F (x, y, q; t) =
X
L(t) =
X
Fa t a ,
a≥1
La t a ,
(8)
Ma t a .
(9)
a≥1
M(t) =
X
a≥1
Clearly, by (3) we have
F (x, y, q) = 1 +
xy
+ L(1) + M(1).
1−x
(10)
Thus substituting (6) into (8) and similarly (7) into (9), we obtain
L(t) =
a−1
XX
x a+j y2 t a +
a≥1 j=1
X
a≥1
xay
a−1
X
j=1
(Lj + qMj ) t a =
∞
∞
X
X
x 3 y2 t 2
+
(Lj + qMj ) y
(tx)a
2
(1 − xt)(1 − x t)
j=1
a=j+1
x yt
xyt
qxyt
+
L(tx) +
M(tx),
(1 − xt)(1 − x 2 t) 1 − tx
1 − tx
3 2 2
=
and
j
∞
X
X
X
xa
x a+j y2 + x a y (qLj + Mj ) t a =
y2 (tx)a
+
y(qLj + Mj )
(tx)a
1
−
x
a≥1
a=1
a≥1 j=a
j=1
∞ j
j
2
X
xyqtLj
xytMj
xyqt (xt) Lj
xyt (xt) Mj
(xy) t
=
+
+
−
−
(1 − x)(1 − tx 2 )
1
−
xt
1
−
xt
1
−
xt
1 − xt
j=1
M(t) =
=
∞
XX
(xy)2 t
xyqt L(1) xyt M(1) xyqt L(xt) xyt M(xt)
+
+
−
−
.
(1 − x)(1 − x 2 t)
1 − xt
1 − xt
1 − xt
1 − xt
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Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions
If we define
x 3 y2 t 2
,
(1 − xt)(1 − x 2 t)
A(t) =
x 2 y2 t
xyt
+
(qL(1) + M(1)),
(1 − x)(1 − x 2 t) 1 − xt
B(t) =
then L(t) and M(t) can be expressed using matrices as follows:
L(t)
M(t)
!
!
xyt
A(t)
+
1 − xt
B(t)
=
1 q
−q −1
!
!
L(xt)
.
M(xt)
(11)
We keep iterating (11) and use the fact that x j , yj , t j → 0 as j → ∞ for |x|, |y|, |t| ≤ ρ < 1. Finally we put t = 1 and
obtain the following result.
Theorem 3.1.
The generating functions L(1) and M(1) are given by
L(1)
M(1)
where
!
X
=
1 q
−q −1
αj
j≥0
!j
x j+2
1−x j+1
x
1−x
!
X
+
βj
j≥0
j+2
x ( 2 ) yj+2
,
(1 − x)(1 − x 2 ) · · · (1 − x j )(1 − x j+2 )
αj =
1 q
−q −1
βj =
!j
!
0
,
qL(1) + M(1)
j+2
x ( 2 ) yj+1
.
(1 − x)(1 − x 2 ) · · · (1 − x j+1 )
Using the fact that
1 q
−q −1
!2j
= 1−q
2 j
I
1 q
−q −1
and
!2j+1
= 1−q
2 j
!
1 q
,
−q −1
Theorem 3.1 gives
L(1)
M(1)
!
X
=
x 2j+2
1−x 2j+1
x
1−x
2 j
α2j 1 − q
j≥0
X
+
β2j 1 − q
qx
x 2j+3
+ 1−x
1−x 2j+2
2j+3
qx
x
− 1−x
2j+2 − 1−x
!
X
+
α2j+1 1 − q
2 j
j≥0
!
X
0
+
β2j+1 1 − q2 j
qL(1) + M(1)
j≥0
2 j
j≥0
!
!
q2 L(1) + qM(1)
.
−qL(1) − M(1)
This is equivalent to

1 − q2
X
β2j+1 1 − q2
j
−q
X
β2j+1 1 − q2
j

 L(1)

 X j≥0
X j≥0

2 j
2 j
q
(β2j+1 − β2j ) 1 − q
1+
(β2j+1 − β2j ) 1 − q
M(1)
!
j≥0
j≥0
X
=
α2j 1 − q2
j
j≥0
x 2j+2
1−x 2j+1
x
1−x
!
X
+
α2j+1 1 − q2
j≥0
j
qx
x 2j+3
+ 1−x
1−x 2j+2
qx 2j+3
x
− 1−x
2j+2 − 1−x
!
.
Since

1 − q2
X
β2j+1 1 − q2
j
−q
X
β2j+1 1 − q2
j
−1


1
 X j≥0
X j≥0
X
j
j  =
T
q
j
(β2j+1 − β2j ) 1 − q2 1 +
(β2j+1 − β2j ) 1 − q2 
1−
(−1) βj 1 − q2 b(j+1)/2c
j≥0
j≥0
j≥0
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with
X

1+
(β2j+1 − β2j ) 1 − q2
j
q
X
β2j+1 1 − q2
j


j≥0
j≥0
X
X
,
2 j
2 j
2
q
(β2j − β2j+1 ) 1 − q
1−q
β2j+1 1 − q

T=

j≥0
j≥0
we obtain
!
L(1)
=
M(1)
T
X
α2j 1 − q2
x 2j+2
1−x 2j+1
x
1−x
j
j≥0
1−
qx
x 2j+3
+ 1−x
1−x 2j+2
qx 2j+3
x
− 1−x
2j+2 − 1−x
!
+T
X
α2j+1 1 − q2
j
j≥0
X
j
(−1) βj 1 − q2
!
.
b(j+1)/2c
(12)
j≥0
We substitute the expressions for L(1) and M(1) into (10) to obtain our main result, which is the generating function
F (x, y, q) as shown in the next theorem.
Theorem 3.2.
Let
G = 1 + (1 − q)
X
(β2j+1 − β2j ) 1 − q2 j ,
H = 1 + q(1 − q)
j≥0
X
β2j+1 1 − q2 j .
j≥0
The generating function F (x, y, q) is given by


X
X
qx
(β2j+1 + β2j+2 ) 1 − q2 j +
G
α2j+1 1 − q2 j 
1 − x j≥0
j≥0
xy
X
+
1+
1−x
1−
(−1)j βj 1 − q2 b(j+1)/2c
j≥0


x
H c 1−x
+
P
j≥0
(α2j − α2j+1 ) 1 − q
2 j
−q
X
β2j+2 1 − q
j≥0
1−
X
(−1)j βj 1 − q2
b(j+1)/2c
2 j

,
j≥0
where
j+2
x ( 2 ) yj+2
αj =
(1 − x)(1 − x 2 ) · · · (1 − x j )(1 − x j+2 )
and
j+2
x ( 2 ) yj+1
βj =
.
(1 − x)(1 − x 2 ) · · · (1 − x j+1 )
The proof is a matter of simplification of the expression for L(1)+M(1), bearing in mind that αj x j+2 /(1−x j+1 ) =
βj+1 . This is immediate from the definitions of αj and βj .
Proof.
Applying the above theorem for q = 1, we obtain G = H = 1 and then
xy
1 x
1 x
+
β1 + β2 +
α1 +
(α0 − α1 ) − β2
1−x
1 − β0
1−x
1 − β0 1 − x
xy
1
x 3 y2
x 2 y2
=1+
+
+
1−x
1 − xy/(1 − x) (1 − x)(1 − x 2 ) (1 − x)(1 − x 2 )
F (x, y, 1) = 1 +
=1+
xy
x 2 y2
1−x
+
=
,
1−x
(1 − x)(1 − x − xy)
1 − x − xy
which is well known, see e.g. [4].
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Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions
We are now in a position to find the generating function for the compositions of n that consist of precisely two alternating
blocks, either starting with an ascending or descending block. Since we are interested only in the occurrence of two
blocks, only one peak or one valley will occur. Thus we need to find the coefficient of q in the generating function in
Theorem 3.2. Hence
Corollary 3.3.
The generating function for compositions of n with two alternating blocks is

X
x
·
1−x
j≥0
β2j+1
X
α2j +
j≥0
X
α2j+1 1 −
j≥0
1−
X

X
X
β2j 
j≥0
−
(−1)j βj
(β2j+1 − β2j )
j≥0
1−
X
X
β2j
j≥1
(−1)j βj
.
j≥0
We leave this proof to the reader, noting that the constant term in
1
1−
X
(−1)j βj 1 − q2
is
b(j+1)/2c
1−
j≥0
4.
βj +
j≥1
j≥0
Proof.
X
1
X
.
(−1)j βj
j≥0
Generating function for the total number of peaks and valleys
In this section, we find the generating function for the total number of peaks and valleys in any composition of n. Let
A=
X
j≥0
B=
qx X
α2j+1 1 − q2 j ,
1 − x j≥0
X
X
(α2j − α2j+1 ) 1 − q2 j − q
β2j+2 1 − q2 j .
(β2j+1 + β2j+2 ) 1 − q2
x
1−x
j
j≥0
+
j≥0
Then Theorem 3.2 gives
d
GA + HB
d
.
X
F (x, y, q) q=1 =
j
2 b(j+1)/2c dq
dq 1 −
(−1) βj 1 − q
q=1
j≥0
Using the facts that
G|q=1 = H|q=1 = 1,
d
A|q=1
dq
A|q=1 = β1 + β2 +
d
G|q=1 = β0 − β1 ,
dq
x
= −2(β3 + β4 ) +
(α1 − 2α3 ),
1−x
x
x
α1 ,
B|q=1 =
(α0 − α1 ) − β2 ,
1−x
1−x
d
H|q=1 = −β1 ,
dq
d
2x
B|q=1 = −
(α2 − α3 ) − β2 + 2β4 ,
dq
1−x
we obtain
d
x α0 (β1 + β1 β0 − 2β2 ) + α1 (1 − β02 ) + 2α2 (β0 − 1)
F (x, y, q)q=1 =
dq
1−x
(1 − β0 )2
−
2β3 + β2 + β1 (2β2 − β1 ) − β0 (2β2 + 2β3 + β12 + β1 ) + β02 (β1 + β2 )
.
(1 − β0 )2
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Theorem 4.1.
The generating function for the total number of peaks and valleys in any composition of n with m parts is given by
(1 −
2x 4 y3
.
− x − xy)2
x 3 )(1
Moreover, the total number of peaks and valleys in any compositions of n with m parts is given by
b(n−3)/3c 2(m − 2)
X
j=0
Proof.
n − 3 − 3j
.
m−2
(13)
The first part follows immediately from substituting the values of αj and βj in (13). Direct calculations show
X 2(m − 2)x m+1
XX
m − 2 + i 3j+i+m+1 m
2x 4 y3
m
=
y
=
2(m
−
2)
x
y ,
(1 − x 3 )(1 − x − xy)2
(1 − x)m−1 (1 − x 3 )
i
m≥2
m≥2 i,j≥0
which implies the total number of peaks and valleys in any compositions of n with m parts is given by (13), as claimed.
Remark.
One can easily obtain the asymptotic expression for the mean number of partition blocks per composition of n, as n
tends to infinity, to be 1/49 + 2n/7. We use the fact that the number of peaks and valleys plus one equals the number
of partition blocks. This result matches the result found in [3].
With our techniques one can obtain more results. For instance, if we denote the generating function for all compositions
of n with m parts where the first part is a descent (respectively, non-descent) by L(x, y, q) (respectively, M(x, y, q)),
then (12) gives the following result.
Corollary 4.2.
The generating functions


L(x, y, q) 1 −
X
(−1)j βj 1 − q
2 b(j+1)/2c
j≥0
,


M(x, y, q) 1 −
X
(−1)j βj 1 − q
2 b(j+1)/2c

j≥0
are given by


X
X
x
q
β2j+1 1 − q 
(α2j − α2j+1 ) 1 − q2 j − q
β2j+2 1 − q2 j 
1 − x j≥0
j≥0
j≥0



X
X
X
qx
+ 1 +
(β2j+1 − β2j ) 1 − q2 j  
βj+1 1 − q2 bj/2c +
α2j+1 1 − q2 j 
1 − x j≥0
j≥0
j≥0
X
2 j
and

X
X
x
1 − q2
(α2j − α2j+1 ) 1 − q2 j − q
β2j+2 1 − q2 j 
β2j+1 1 − q  
1
−
x
j≥0
j≥0
j≥0


X
X
X
qx
+q
(β2j − β2j+1 ) 1 − q2 j 
βj+1 1 − q2 bj/2c +
α2j+1 1 − q2 j  ,
1
−
x
j≥0
j≥0
j≥0


X
2 j
respectively.
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Ars Combin. (in press)
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Boca Raton, 2010
[5] MacMahon P., Combinatory Analysis, Cambridge University Press, Cambridge, 1915–1916, reprinted by Chelsea,
New York, 1960
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1997
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