The combinatorics of MacMahon's partial fractions

TL;DR

This paper generalizes MacMahon's partial fractions decomposition for generating functions of partitions, offering a comprehensive combinatorial explanation and extending the classical result.

Contribution

It introduces a generalized form of MacMahon's partial fractions decomposition with a full combinatorial interpretation.

Findings

Generalization of MacMahon's partial fractions for partition generating functions

Provides a combinatorial explanation for the decomposition

Extends classical results to broader partition contexts

Abstract

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation.

Tables

Table 1. Table 1. The letters a , b , c , d 𝑎 𝑏 𝑐 𝑑 a,b,c,d represent distinct nonnegative integers. Permutations are presented in cycle notation.

$\lambda$	Form of $\gamma$	Generating terms of RHS of (4.1)
	$aaaa$	$6g(4;𝐱)+2\left(\left(\right)+\left(34\right)+\left(24\right)+\left(14\right)\right)g(31;𝐱)$
$\left(4\right)$		$+\left(\left(\right)+\left(23\right)+\left(24\right)\right)g(22;𝐱)$
		$+\left(\left(\right)+\left(13\right)+\left(23\right)+\left(24\right)+\left(14\right)+\left(13\right)\left(24\right)\right)g(211;𝐱)$
		$+g(1111;𝐱)$
	$aaab$	$2g(31;𝐱)+\left(\left(\right)+\left(13\right)+\left(23\right)\right)g(211;𝐱)+g(1111;𝐱)$
$\left(31\right)$	$aaba$	$2\left(34\right)g(31;𝐱)+\left(\left(\right)+\left(24\right)+\left(14\right)\right)g(211;𝐱)+g(1111;𝐱)$
	$abaa$	$2\left(24\right)g(31;𝐱)+\left(\left(23\right)+\left(24\right)+\left(13\right)\left(24\right)\right)g(211;𝐱)+g(1111;𝐱)$
	$abbb$	$2\left(14\right)g(31;𝐱)+\left(\left(13\right)+\left(14\right)+\left(13\right)\left(24\right)\right)g(211;𝐱)+g(1111;𝐱)$
	$aabb$	$g(22;𝐱)+\left(\left(\right)+\left(13\right)\left(24\right)\right)g(211;𝐱)+g(1111;𝐱)$
$\left(22\right)$	$abab$	$\left(23\right)g(22;𝐱)+\left(\left(23\right)+\left(14\right)\right)g(211;𝐱)+g(1111;𝐱)$
	$abba$	$\left(24\right)g(22;𝐱)+\left(\left(13\right)+\left(24\right)\right)g(211;𝐱)+g(1111;𝐱)$
	$aabc$	$g(211;𝐱)+g(1111;𝐱)$
	$abac$	$\left(23\right)g(211;𝐱)+g(1111;𝐱)$
$\left(211\right)$	$abca$	$\left(24\right)g(211;𝐱)+g(1111;𝐱)$
	$abbc$	$\left(13\right)g(211;𝐱)+g(1111;𝐱)$
	$abcb$	$\left(14\right)g(211;𝐱)+g(1111;𝐱)$
	$abcc$	$\left(13\right)\left(24\right)g(211;𝐱)+g(1111;𝐱)$
$\left(1111\right)$	$abcd$	$g(1111;𝐱)$

Equations

$$\frac{1}{(1-x)(1-x^2)}$$

$$\frac{1}{(1-x)(1-x^2)(1-x^3)}$$

$$\frac{1}{(1)(2)}$$

$$\frac{1}{(1)(2)(3)}$$

$$\frac{1}{(1)(2)\cdots (\mathbf{i})}=\sum \frac{1}{1^{p_1}.2^{p_2}.3^{p_3}\dots p_1!p_2!p_3!\dots }\frac{1}{{(1)}^{p_1}{(2)}^{p_2}{(3)}^{p_3}\dots },$$

$$\frac{1}{1^{p_1}.2^{p_2}.3^{p_3}\dots p_1!p_2!p_3!\dots }.$$

$$f_j(\gamma ):=\#\{i:{\gamma }_i=j\}.$$

$$f(\gamma ):=(f_0(\gamma ),f_1(\gamma ),f_2(\gamma ),f_3(\gamma ),\dots ).$$

$$w_1\ge w_2\ge \cdots \ge w_k.$$

$$f(\lambda )=(f_1(\lambda ),f_2(\lambda ),f_3(\lambda ),\dots ).$$

$$\mathcal{P}(4)=\{(4),(3,1),(2,2),(2,1,1),(1,1,1,1)\},$$

$$\lambda -1:=({\lambda }_1-1,{\lambda }_2-1,\dots ,{\lambda }_{l(\lambda )}-1).$$

$$(5,5,5,5,3,2,2,1,1,1)=⟨1^3{2}^{2}35^4⟩$$

$$N\left(\lambda \right):={\lambda }_1{\lambda }_2\cdots {\lambda }_k.$$

$$N\left(\gamma !\right):={\gamma }_1!{\gamma }_2!\dots {\gamma }_k!$$

$$N\big{(}f(\gamma)!\big{)}=f_{0}(\gamma)!f_{1}(\gamma)!f_{2}(\gamma)!\cdots,$$

$$N\big{(}f(\lambda)!\big{)}=f_{1}(\lambda)!f_{2}(\lambda)!f_{3}(\lambda)!\cdots.$$

$$MD\left(\lambda \right):=\mathcal{P}({\lambda }_1)\times \mathcal{P}({\lambda }_2)\times \cdots \times \mathcal{P}({\lambda }_{l(\lambda )}).$$

$$\sum _{\lambda ⊢n}\frac{1}{1^{f_1}{2}^{f_2}{3}^{f_3}\cdots f_1!f_2!f_3!\cdots }=1,$$

$$\sum_{\lambda\vdash n}\frac{1}{N\left(\lambda\right)N\big{(}f(\lambda)!\big{)}}=1,$$

$$\sum_{i=1}^{p(\lambda_{1})p(\lambda_{2})\cdots p(\lambda_{l(\lambda)})}\prod_{j=1}^{l(\lambda)}\frac{1}{N\big{(}f(\mu^{ij})!\big{)}N\left(\mu^{ij}\right)}=1.$$

$$\frac{1}{(1-x)(1-x^2)\cdots (1-x^k)}=\sum _{\lambda ⊢k}\frac{1}{1^{f_1}{2}^{f_2}{3}^{f_3}\cdots f_1!f_2!f_3!\cdots }g(\lambda ;x,x,\dots ,x),$$

$$g(\lambda ;\mathbf{x}):=g(\lambda ;x_1,x_2,\dots ,x_{|\lambda |}):=\prod _{j=1}^{l(\lambda )}\frac{1}{1-{\prod }_{k=1}^{{\lambda }_j}{x}_{s(\lambda ;j,k)}},$$

$$s(\lambda ;j,k)=k+\sum _{r=1}^{j-1}{\lambda }_r.$$

$$\displaystyle g\Big{(}(4);x_{1},x_{2},x_{3},x_{4}\Big{)}$$

$$\displaystyle g\Big{(}(3,1);x_{1},x_{2},x_{3},x_{4}\Big{)}$$

$$\displaystyle g\Big{(}(2,2);x_{1},x_{2},x_{3},x_{4}\Big{)}$$

$$\displaystyle g\Big{(}(2,1,1);x_{1},x_{2},x_{3},x_{4}\Big{)}$$

$$\displaystyle g\Big{(}(1,1,1,1);x_{1},x_{2},x_{3},x_{4}\Big{)}$$

$$\sigma g(\lambda ;\mathbf{x})=g(\lambda ;\sigma \mathbf{x})=g(\lambda ;x_{\sigma (1)},x_{\sigma (2)},\dots ,x_{\sigma (|\lambda |)}).$$

$$\left|H_{\lambda}\right|=N\left(\lambda!\right)N\big{(}f(\lambda)!\big{)},$$

$$\left|\mathscr{O}_{\lambda}\right|=\frac{|\lambda|!}{N\left(\lambda!\right)N\big{(}f(\lambda)!\big{)}}.$$

$$N\left((\lambda-\mathbf{1})!\right)|\mathscr{O}_{\lambda}|=\frac{|\lambda|!}{N\left(\lambda\right)N\big{(}f(\lambda)!\big{)}},$$

Full text

The author thanks the National Security Agency for partially supporting his research program via grant H98230-14-1-0159 during 2014–2015, when this research project commenced.

Dedicated to my teacher, George E. Andrews, on the occasion of his 80th birthday

1. Introduction

1.1. An excerpt from MacMahon’s Combinatory Analysis

In Combinatory Analysis, vol. 2 [7, p. 61ff], P. A. MacMahon writes:

We commence by observing the two identities

[TABLE]

which we will also write in the illuminating notation so often employed:

[TABLE]

[Elsewhere [7, p. 5] the “very convenient notation” $\mathbf{(j)}=1-x^{j}$ is defined and attributed to Cayley.]

The observation leads to the conjecture that we are in the presence of partial fractions of a new and special kind. We note that in the first identity we have a fraction corresponding to each of the partitions $(1^{2})$, $(2)$ of the number $2$ and in the second fractions corresponding to and derived from each of the partitions $(1^{3})$, $(21)$, $(3)$ of the number $3$. …[In general] we find

[TABLE]

where $(1^{p_{1}}2^{p_{2}}3^{p_{3}}\dots)$ is a partition of $i$ and the summation is in regard to all partitions of $i$. This remarkable result shows the decomposition of the generating function into as many fractions as the number $i$ possesses partitions. The denominator of each fraction is directly derived from one of the partitions and is of degree $i$ in $x$. The numerator does not involve $x$ and the coefficient is the easily calculable number

[TABLE]

Remark 1.1**.**

In [3, p. 209, Ex. 1] Andrews attributes (1.1) to Cayley, and this attribution has been repeated by other authors in the literature. However, the author has been unable to find (1.1) anywhere in Cayley’s works [5], and indeed MacMahon, in the chapter where he presents his “partial fractions of a new and special kind” [7, Section VII, Chapter V] contrasts his results with those of Cayley several times.

1.2. Some definitions and notation

1.2.1. Partitions and related objects

It will be necessary to employ partitions and compositions of positive integers, sometimes allowing [math]’s as parts, and sometimes not. Accordingly, we will formalize terminology via the following definitions.

Definition 1.2**.**

A weak $k$-composition $\gamma$ is a $k$-tuple of nonnegative integers $(\gamma_{1},\gamma_{2},\dots,\gamma_{k})$. Each $\gamma_{i}$ (even if $\gamma_{i}=0$) is called a part of $\gamma$. The weight of $\gamma$, denoted $|\gamma|$, is $\sum_{i=1}^{m}\gamma_{i}$. The length of $\gamma$, denoted $l(\gamma)$, is the number of parts in $\gamma$. The frequency (or multiplicity) of part $j$ in $\gamma$, denoted $f_{j}(\gamma)$ or simply $f_{j}$ when $\gamma$ is clear from context, is the number of times that $j$ appears as a part in $\gamma$:

[TABLE]

The frequency sequence associated with $\gamma$ is

[TABLE]

The set of all weak $k$-compositions will be denoted $\mathscr{C}_{k}$.

Remark 1.3**.**

The author prefers to use the term frequency and corresponding notation $f_{j}$ over the term multiplicity (with notation $m_{j}$) to be consistent with the works of Andrews [1, 2, 3].

Definition 1.4**.**

A weak $k$-composition $w=(w_{1},w_{2},\dots,w_{k})$ is a weak $k$-partition if its parts occur in nonincreasing order

[TABLE]

The set of weak $k$-partitions of weight $n$ will be denoted $\mathscr{W}_{k}(n)$ and the cardinality of this set by $p_{k}(n)$.

Definition 1.5**.**

If $w$ is a weak $k$-partition and $\gamma$ is a weak $k$-composition, we shall say that $\gamma$ is of type $w$ if $\gamma$ is a permutation of $w$.

Definition 1.6**.**

A partition $\lambda$ is any nonincreasing finite or infinite sequence $(\lambda_{1},\lambda_{2},\lambda_{3},\dots)$ of nonnegative integers. However, in contrast to Definition 1.2, only positive integers are considered parts, thus for a partition $\lambda$, $l(\lambda)=\#\{i:\lambda_{i}>0\}.$ Analogous to the frequency sequence of a weak composition, the frequency sequence $f(\lambda)$ of $\lambda$ is

[TABLE]

Remark 1.7**.**

In fact, no distinction will be drawn between, e.g., $\lambda=(5,2,1,1)$ and $\lambda=(5,2,1,1,0,0,0,0,0,0,0,0,\dots)$; both will be considered the same partition of length $4$ and weight $9$. Also, f\Big{(}(5,2,1,1)\Big{)}=(2,1,0,0,1,0,0,\dots).

It will be convenient to consider $\lambda_{j}=0$ for any $j>l(\lambda)$, even when $\lambda$ is not explicitly constructed with a tail of zeros.

Definition 1.8**.**

The set of all partitions of weight $n$ is denoted by $\mathscr{P}(n)$ and the cardinality of $\mathscr{P}(n)$ by $p(n)$. The notation $\lambda\vdash n$ means “$\lambda$ is a partition of weight $n$”, i.e., $\lambda\in\mathscr{P}(n)$.

For example,

[TABLE]

so $p(4)=5$.

Definition 1.9**.**

For a partition $\lambda$, the partition $\lambda-\mathbf{1}$ is the partition obtained from $\lambda$ by decreasing each of its parts by $1$:

[TABLE]

Notice that $l(\lambda-\mathbf{1})=l(\lambda)-f_{1}(\lambda).$

It is often convenient to denote a partition (resp. weak $k$-partition) by the superscript frequency notation $\langle 1^{f_{1}}2^{f_{2}}3^{f_{3}}\cdots\rangle$ (resp. $\langle 0^{f_{0}}1^{f_{1}}2^{f_{2}}3^{f_{3}}\cdots\rangle$) where it is permissible to omit $f_{j}$ if $f_{j}=1$ and to omit $j^{f_{j}}$ if $f_{j}=0$. Thus,

[TABLE]

are two ways of expressing that particular (weak $10$-)partition of $30$.

A variant on this notation for weak compositions (in order to emphasize runs of adjacent equal parts) will also be useful. For example, let us allow ourselves to write the weak $9$-composition $(3,3,2,0,0,3,3,3,3)$ of $20$ as $[3^{2}2^{1}0^{2}3^{4}]$.

The following quantities will arise often enough to warrant these definitions:

Definition 1.10**.**

Following Schneider [11], for a partition $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{k})$, we define its norm to be the product of its parts,

[TABLE]

Further, the factorial of a partition $\lambda$ is $\lambda!=(\lambda_{1}!,\lambda_{2}!,\dots,\lambda_{k}!)$, so that $N\left(\lambda!\right)=\lambda_{1}!\lambda_{2}!\cdots\lambda_{k}!.$ Analogously for a weak $k$-composition $\gamma=(\gamma_{1},\dots,\gamma_{k})$,

[TABLE]

Observe that $f$ effectively maps a partition $\lambda=(\lambda_{1},\lambda_{2},\dots)$ to a weak $\lambda_{1}$-composition of weight $l(\lambda)$ if we ignore the infinite tail of zeros in $f(\lambda)$. For example, $f((5,5,5,5,3,2,2,1,1,1))=(3,2,1,0,4)$, a weak $5$-composition of weight $10$. Likewise, $f$ maps a weak $k$-composition $\gamma$ to a weak $L$-composition of weight $k$, where $L$ is the largest part of $\gamma$. Thus the frequency factorial product of a weak $k$-composition $\gamma$ may be consistently notated as

[TABLE]

and that of a partition $\lambda$ as

[TABLE]

Observation 1.11**.**

The number of weak $k$-compositions of type $w=(w_{1},\dots,w_{k})$, where $w$ is a weak $k$-partition, is k!/N\big{(}f(w)!\big{)}.

Definition 1.12**.**

A multipartition is a $t$-tuple of partitions for some $t$.

For example, $((1,1,1),(4,1),(3,2,2))$ is a multipartition simply because $(1,1,1)$, $(4,1)$, and $(3,2,2)$ are all partitions.

Definition 1.13**.**

The multipartition dissection $M\!D\left(\lambda\right)$ of the partition $\lambda$ is the following Cartesian product:

[TABLE]

We will require the result given by N. J. Fine [6, p. 38, Eq. (22.2)],

[TABLE]

which may be expressed in the present notation as

[TABLE]

in the iterated form

[TABLE]

The superscript notation on $\mu$ is to be understood as follows: $\lambda=(\lambda_{1},\dots,\lambda_{l})$ is a partition; then $\mu^{ij}$ is the $i$th partition of $\lambda_{j}$ where the $p(\lambda_{1})p(\lambda_{2})\cdots p(\lambda_{l})$ multipartitions of $M\!D\left(\lambda\right)$ have been placed in some order; any order is fine. See also (3.3) below for an explicit illustration.

Notice that (1.2) states that the sum of the coëfficients that appear in the MacMahon partial fractions decomposition

[TABLE]

where $g$ is defined below in Eq. (1.5), must be $1$.

1.2.2. Combinatorial Generating Functions

As part of the combinatorial construction to be undertaken, we will need to associate with each partition $\lambda$ a certain rational generating function in indeterminates $x_{1},x_{2},\dots,x_{|\lambda|}$; namely let

[TABLE]

with

[TABLE]

Of necessity, the notation used in defining (1.5) for a general partition $\lambda$ makes a simple idea rather opaque. To understand immediately how to construct $g(\lambda;\mathbf{x})$ for any partition $\lambda$, simply consider, for example, for the five partitions of $4$, we have the following associated “$g$-functions”:

[TABLE]

We denote the symmetric group of degree $n$ by $\mathfrak{S}_{n}$. The application of a permutation $\sigma\in\mathfrak{S}_{|\lambda|}$ to $g(\lambda;\mathbf{x})$, will be written as $\sigma g(\lambda;\mathbf{x})$, with the intended meaning

[TABLE]

Let $\mathscr{O}_{\lambda}$ (resp. $H_{\lambda}$) denote the orbit (resp. stabilizer) of $g(\lambda;\mathbf{x})$ under the action of $\mathfrak{S}_{|\lambda|}$.

Thus,

[TABLE]

or, by the orbit–stabilizer theorem,

[TABLE]

Remark 1.14**.**

Notice that

[TABLE]

which is $|\lambda|!$ times the coëfficient of the term indexed by $\lambda$ in the MacMahon decomposition.

1.3. Statement of main result

The goal is to understand (1.1) combinatorially. This will be accomplished by proving the following natural multivariate generalization of (1.1):

Theorem 1.15**.**

[TABLE]

where $\mathscr{O}_{\lambda}$ is the orbit of $g(\lambda;\mathbf{x})$ under the action of $\mathfrak{S}_{|\lambda|}$, and $g(\lambda;\mathbf{x})$ is defined in (1.5).

2. Partial Fractions Decompositions

It is well known that for fixed positive integer $k$, the generating function for $p_{k}(n)$ is

[TABLE]

Since the right-hand side of (2.1) is a rational function, it can be decomposed into ordinary partial fractions, as considered, e.g, by Cayley [4] and Rademacher [10, p. 302], or into $q$-partial fractions, as studied by Munagi [8, 9].

In examining the ordinary partial fraction decompositions of, say, $F_{4}(x)$,

[TABLE]

where $\omega:=\exp(2\pi i/3),$ we notice immediately the apparent arbitrariness of the coëfficients that arise in the expansion.

For Munagi’s $q$-partial fractions, the coëfficients are nicer, but still not transparent:

[TABLE]

MacMahon’s partial fraction decomposition of $F_{k}(x)$,

[TABLE]

thus has the distinct advantage that the coëfficients are known a priori, and furthermore these coëfficients are “combinatorial numbers” in the sense that they are products of integer exponential and factorial expressions.

In order to begin to understand (2.5) combinatorially, we shall multiply both sides of (2.5) by $k!$ and observe that $g(\lambda;x,x,x,\dots,x)$ is the generating function for the sequence that counts a certain class of restricted weak $k$-compositions defined below.

Equation (2.5) together with (1.7), after some investigation, suggested the generalization of MacMahon’s partial fraction decomposition presented above as Theorem 1.15.

3. Proof of Theorem 1.15

Starting with the left member of (1.8), we have

[TABLE]

Thus, we see that the left member of (1.8) generates every weak $k$-composition (where the $j$th part appears as the exponent of $x_{j}$) exactly N\big{(}f(\gamma)!\big{)} times.

Now let us consider the right member of (1.8),

[TABLE]

where $\mathscr{O}_{\lambda}$ is the orbit of $g(\lambda;\mathbf{x})$ under the (transitive) action of $\mathfrak{S}_{|\lambda|}$.

Pick an arbitrary weak $k$-composition $\gamma$. We need to show that the term $x_{1}^{\gamma_{1}}x_{2}^{\gamma_{2}}\cdots x_{k}^{\gamma_{k}}$ appears in the expansion of (3.1) with coëfficient N\big{(}f(\gamma)!\big{)}. Associated with $\gamma$ is the frequency sequence $f(\gamma)=(f_{0}(\gamma),f_{1}(\gamma),f_{2}(\gamma),\dots)$. Permute the nonzero terms of $f(\gamma)$ into nondecreasing order to form a partition $\lambda$ of weight $k$, and we may write $\lambda=\lambda(\gamma)$, since the partition $\lambda$ is uniquely determined by $\gamma$. Thus it must be the case that there exists $\sigma\in\mathfrak{S}_{k}$ such that the weak $k$-composition $\sigma(\gamma)$ is of type $[c_{1}^{\lambda_{1}}c_{2}^{\lambda_{2}}\cdots c_{l}^{\lambda_{l}}]$ for some distinct nonnegative integers $c_{1},c_{2},\dots,c_{l}$.

For a given $\lambda\vdash k$ of length $l$, we have, by expanding (1.5) as a series,

[TABLE]

so $g(\lambda;\mathbf{x})$ is the generating function for weak $k$-compositions of type

[TABLE]

Now the orbit $\mathscr{O}_{\lambda}$ of $g(\lambda;\mathbf{x})$ under the action of $\mathfrak{S}_{k}$ contains the terms that generate all permutations of weak $k$-compositions of type $[c_{1}^{\lambda_{1}}c_{2}^{\lambda_{2}}\cdots c_{l}^{\lambda_{l}}]$.

The terms $x_{1}^{\gamma_{1}}x_{2}^{\gamma_{2}}\cdots x_{k}^{\gamma_{k}}$ are generated by those terms of (3.1) in the orbit of $g(\mu;\mathbf{x})$ for multipartitions $\mu\in M\!D\left(\lambda\right)$ where $\lambda=\lambda(\gamma)$.

For each weak $k$-composition $\gamma$, and corresponding partition $\lambda=\lambda(\gamma)$ $=(\lambda_{1},\lambda_{2},\dots,\lambda_{l(\lambda)})$, we generate all the associated multipartitions in $M\!D\left(\lambda\right)$. Let $\mu^{ij}_{k}$ denote the $k$th part in the partition $\mu^{ij}$, where $\mu^{ij}$ is the $i$th partition of $\lambda_{j}$, the $j$th part of $\lambda$.

Note that $i$ runs from $1$ through $p(\lambda_{1})p(\lambda_{2})\cdots p(\lambda_{l(\lambda)})$ where some ordering has been imposed on the multipartitions (any ordering will do). Of course, $j$ runs from $1$ to $l(\lambda)$, and $k$ runs from $1$ to $l(\mu^{ij})$.

For example, if we wish to calculate the number of times the weak $5$-composition $\gamma=(7,7,4,7,4)$ is generated by the right-hand side of (1.8), i.e., the number of times the expression $x_{1}^{7}x_{2}^{7}x_{3}^{4}x_{4}^{7}x_{5}^{4}$ appears, we see, by symmetry, that this must be the same as the number of times $(7,7,7,4,4)$ appears.

The weak $5$-partition $(7,7,7,4,4)$ is clearly obtained from $(7,7,4,7,4)$ by allowing the permutation $\sigma=(3,4)$ to act on it. Then $\lambda(\gamma)=\lambda(\sigma\gamma)=(3,2)$ because $\gamma$ (and $\sigma\gamma$) both contain $3$ of one part and $2$ of another part.

We notice that a certain number of copies of $(7,7,7,4,4)$ are generated by each of the terms

[TABLE]

and by no other terms. To aid our analysis we consider the multipartition dissection of the partition $\lambda=(3,2)$:

[TABLE]

because each of these six multipartitions indexes a term that generates some number of copies of $\sigma\gamma=(7,7,7,4,4).$ (In this example, we have $\mu^{1}=((3),(2))$, $\mu^{2}=((3),(1,1))$, $\mu^{3}=((2,1),(2))$, $\mu^{4}=((2,1),(1,1))$, $\mu^{5}=((1,1,1),(2))$, and $\mu^{6}=((1,1,1),(1,1))$; $\mu^{11}=\mu^{21}=(3)$, $\mu^{12}=\mu^{32}=\mu^{52}=(2)$, $\mu^{22}=\mu^{42}=\mu^{62}=(1,1)$, and $\mu^{51}=\mu^{61}=(1,1,1)$.)

We use elementary combinatorial reasoning to count how many copies of $(7,7,7,4,4)$ are generated by each of the six terms. That number is a consequence of the commutivity of ordinary multiplication. For example, consider the third listed term in (3.2)

To generate $(7,7,7,4,4)$, we may do so by any of the following permutations of this third term:

[TABLE]

which are indexed by the multipartition $\mu^{3}=((2,1),(2))$.

This clearly lists all elements in the Cartesian product of the two orbits: one is the orbit of $\frac{1}{(1-x_{1}x_{2})(1-x_{3})}$ under the action of $\mathfrak{S}_{3}=\mathfrak{S}_{\{1,2,3\}}$, (the permutations of $\{1,2,3\}$), and the other is the orbit of $\frac{1}{1-x_{4}x_{5}}$ under the action of $\mathfrak{S}_{\{4,5\}}$ (the permutations of $\{4,5\}$). Since each term generates one copy of $(7,7,7,4,4)$, the total contribution of these terms is given by

[TABLE]

where we have applied Remark 1.14.

Of course, to generate all copies of $(7,7,7,4,4)$, we must sum over all of the terms indexed by the six members of $M\!D\left((3,2)\right)$, employing the analogous counting formula in each case.

In the general case, the preceding combinatorial argument yields

[TABLE]

Thus all that remains in order to prove Theorem 1.15 is to establish:

[TABLE]

Since N\big{(}f(\gamma)!\big{)}=N\left(\lambda!\right)=\prod_{j}\lambda_{j}!, we immediately see that (3.5) is equivalent to the assertion

[TABLE]

which is exactly (1.4), and thus Theorem 1.15 is established. ∎

4. Example: the case $k=4$

Before concluding, let us examine the $k=4$ case in some detail. Our main result, Theorem 1.15, in the case $k=4$ asserts

[TABLE]

In the left member of (4.1), we have

[TABLE]

which generates every weak $4$-partition $w=(w_{1},w_{2},w_{3},w_{4})$ exactly once. The cardinality of the orbit of the action of $\mathfrak{S}_{4}$ on $w$ is

[TABLE]

i.e., there are $4!/(f_{0}(w)!f_{1}(w)!f_{2}(w)!\cdots)$ distinct weak $4$-compositions of type $w$. Or equivalently, a given weak $4$-compositon $\gamma$ which equals $\sigma w$ for some permutation $\sigma\in\mathfrak{S}_{4}$, is generated N\big{(}f(\gamma)!\big{)}=f_{0}(\gamma)!f_{1}(\gamma)!f_{2}(\gamma)!\cdots times.

The generation of weak $4$-compositions on the right side of (4.1) is more subtle. Notice that the terms of the right side are grouped according to the partitions of $4$ (which index the sum on the right side) in the order $(4)$, $(3,1)$, $(2,2)$, $(2,1,1)$, $(1,1,1,1)$. For a given weak $4$-composition $\gamma$, the multiplicities of the parts determine which of the terms of the right side contribute to its generation.

A detailed summary is provided in Table 1. In order to make sure the table is clear, let us look at one line of it in detail. Observe the case with $\lambda=(2,2)$ and form of $\gamma$ as $abab$. The $abab$ means we are considering weak $4$-compositions where the first and third parts are the same, and the second and fourth parts are the same, but the first and second parts are different. The corresponding terms from the right member of (4.1) are equivalent to

[TABLE]

The first term of (4.2) is $(23)g(22,\mathbf{x})$, i.e. apply the transposition $(23)$ to

[TABLE]

this yields

[TABLE]

Expand each factor of the right side of (4.3) as a geometric series to find that weak $4$-compositions $(w_{1},w_{2},w_{3},w_{4})$ are generated (in the exponents of the $x_{i}$’s) in which $w_{1}=w_{3}$ and $w_{2}=w_{4}$, i.e., compositions of the type $(a,b,a,b)$. Is this the only way that compositions of type $(a,b,a,b)$ may be generated? No. Consider the second term of (4.2), $(23)g(211,\mathbf{x})$, which is

[TABLE]

This term generates weak $4$-compositions $(w_{1},w_{2},w_{3},w_{4})$, in which $w_{1}=w_{3}$. Some of the weak compositions generated by this term will happen to have $w_{2}=w_{4}$, and thus these will be of the form $(a,b,a,b)$ as well, i.e., this term generates compositions of the general form $(a,b,a,c)$; on those occasions that it happens to be the case that $b=c$, we have a weak composition of the form considered by this particular line of the table. And so on, with the third and forth terms of  (4.2). The other lines of the table may be interpreted similarly.

Acknowledgments

The author thanks George Andrews for pointing out [3, p. 209, Ex. 1], which lead to the research culminating in this paper. The author thanks Matthew Katz for his interest and useful suggestions. The author particularly thanks Robert Schneider for discussions and encouragement of this project over a long period of time, and for carefully reading and offering concrete suggestions to improve earlier versions of the manuscript. Finally, the author is extremely grateful to the editor and anonymous referees for carefully reading the manuscript, catching errors, offering numerous helpful suggestions, and for their kind patience as the author prepared revisions.