Analysis and combinatorics of partition zeta functions

TL;DR

This paper studies partition zeta functions, providing explicit formulas, analyzing their analytic properties, and offering combinatorial proofs, thereby extending the understanding of these functions analogous to the Riemann zeta function.

Contribution

It introduces explicit formulas and analytic insights for a family of partition zeta functions, including a combinatorial proof linking to MacMahon's partial fractions.

Findings

Explicit formulas for partition zeta functions of fixed length

Complete analysis of poles and roots of these functions

A combinatorial proof connecting to MacMahon's decomposition

Abstract

We examine "partition zeta functions" analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties -- those summed over partitions of fixed length -- which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon's partial fraction decomposition of the generating function for partitions of fixed length.

Equations

$${\zeta }_{{\mathcal{P}}^{\prime }}(s):=\sum _{\lambda \in {\mathcal{P}}^{\prime }}N(\lambda {)}^{-s}.$$

$${\zeta }_{{\mathcal{P}}_S}(s)=\prod _{n\in S}{\left(1-n^{-s}\right)}^{-1},$$

$${\zeta }_{{\mathcal{P}}_{\text{even}}}(2)=\frac{\pi }{2},{\zeta }_{{\mathcal{P}}_{\text{dist}}}(2)=\frac{\sinh \pi }{\pi },{\zeta }_{{\mathcal{P}}_{\ne 1}}(3)=\frac{3\pi }{\cosh \left(\frac{1}{2}\pi \sqrt{3}\right)}.$$

$$\zeta (2m)={\pi }^{2m}\times \text{rational number?}$$

$${\zeta }_{\mathcal{P}}(\{s{\}}^k):=\sum _{\ell (\lambda )=k}\frac{1}{N(\lambda {)}^s},$$

$$\prod _{n=1}^{\infty }{\left(1-z{n}^{-s}\right)}^{-1}=\sum _{k=0}^{\infty }{\zeta }_{\mathcal{P}}(\{s{\}}^k)z^k.$$

$${\zeta }_{\mathcal{P}}(\{2{\}}^k)=\frac{2^{2k-1}-1}{2^{2k-2}}\zeta (2k).$$

$${\zeta }_{\mathcal{P}}(\{2m{\}}^k)={\pi }^{2mk}\times \text{rational number}.$$

$${\zeta }_{\mathcal{P}}(\{s{\}}^k)=\sum _{\lambda ⊢k}\frac{\zeta (s{)}^{m_1}\zeta (2s{)}^{m_2}\zeta (3s{)}^{m_3}\cdots \zeta (ks{)}^{m_k}}{N(\lambda )m_1!m_2!m_3!\cdots m_k!},$$

$$-\log (1-x)=\sum _{j=1}^{\infty }\frac{x^j}{j},$$

$$\exp \left(\sum _{j=1}^{\infty }{a}_j{x}^j\right)=\sum _{\lambda \in \mathcal{P}}{x}^{|\lambda |}\frac{a_1^{m_1}{a}_2^{m_2}{a}_3^{m_3}\cdots }{m_1!m_2!m_3!\cdots }=\sum _{k=0}^{\infty }{x}^k\sum _{\lambda ⊢k}\frac{a_1^{m_1}{a}_2^{m_2}{a}_3^{m_3}\cdots a_k^{m_k}}{m_1!m_2!m_3!\cdots m_k!}.$$

$$(a_1+a_2+a_3+\cdots +a_k{)}^n=n!\sum _{\begin{array}{c}\lambda \in {\mathcal{P}}_{[k]}\\ \ell (\lambda )=n\end{array}}\frac{a_1^{m_1}{a}_2^{m_2}{a}_3^{m_3}...a_k^{m_k}}{m_1!m_2!m_3!...m_k!}.$$

$$\exp (a_1+a_2+a_3+\cdots )=\sum _{\lambda \in \mathcal{P}}\frac{a_1^{m_1}{a}_2^{m_2}{a}_3^{m_3}...}{m_1!m_2!m_3!...}.$$

$$\prod _{n=1}^{\infty }\text{exp}\left(-\log \left(1-z{n}^{-s}\right)\right)=\text{exp}\left(\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }\frac{z^j}{n^{js}j}\right)=\text{exp}\left(\sum _{j=1}^{\infty }\frac{\zeta (js)}{j}{z}^j\right).$$

$$\zeta (s)=\sum _{n\ge 1}{n}^{-s}={\zeta }_{\mathcal{P}}(\{s{\}}^1).$$

$$\sum _{\lambda ⊢2}\frac{\zeta (s{)}^{m_1}\zeta (2s{)}^{m_2}}{N(\lambda )m_1!m_2!}=\frac{1}{2}\zeta (s{)}^2+\frac{1}{2}\zeta (2s)$$

$$=\frac{1}{2}\sum _{n_1\ge 1}{n}_1^{-s}\sum _{n_2\ge 1}{n}_2^{-s}+\frac{1}{2}\sum _{n\ge 1}(n^2{)}^{-s}$$

$$=\frac{1}{2}\sum _{(n_1,n_2)\in {\mathcal{C}}_2}\frac{1}{(n_1{n}_2{)}^s}+\frac{1}{2}\underset{n_1=n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}$$

$$=\frac{1}{2}\left(\underset{n_1\ne n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}+\underset{n_1=n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}\right)+\frac{1}{2}\underset{n_1=n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}$$

$$=\frac{1}{2}\underset{n_1\ne n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}+\left(\frac{1}{2}+\frac{1}{2}\right)\underset{n_1=n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}$$

$$=\frac{1}{2}\underset{n_1\lessgtr n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}+\underset{n_1=n_2}{\sum _{(n_1,n_2)\in {\mathcal{C}}_2}}\frac{1}{(n_1{n}_2{)}^s}$$

$$=\sum _{n_1>n_2\ge 1}\frac{1}{(n_1{n}_2{)}^s}+\sum _{n_1=n_2\ge 1}\frac{1}{(n_1{n}_2{)}^s}$$

$$=\sum _{\begin{array}{c}\pi \in \mathcal{P}\\ \ell (\pi )=2\end{array}}\frac{1}{N(\pi {)}^s}={\zeta }_{\mathcal{P}}(\{s{\}}^2).$$

$$\displaystyle\sum_{\lambda\vdash 3}\frac{\zeta(s)^{m_{1}}\zeta(2s)^{m_{2}}\zeta(3s)^{m_{3}}}{N(\lambda)m_{1}!m_{2}!m_{3}!}=\frac{1}{6}\Big{(}\zeta(s)\Big{)}^{3}+\frac{1}{2}\zeta(2s)\zeta(s)+\frac{1}{3}\zeta(3s)$$

$$=\frac{1}{6}\sum _{n_1\ge 1}{n}_1^{-s}\sum _{n_2\ge 1}{n}_2^{-s}\sum _{n_3\ge 1}{n}_3^{-s}+\frac{1}{2}\sum _{n_1\ge 1}(n_1^2{)}^{-s}\sum _{n_2\ge 1}{n}_2^{-s}+\frac{1}{3}\sum _{n\ge 1}(n^3{)}^{-s}$$

$$=[\frac{1}{6}\sum _{(n_1,n_2,n_3)\in {\mathcal{C}}_3}+\left(\frac{1}{6}\underset{n_1=n_2}{\sum _{(n_1,n_2,n_3)\in {\mathcal{C}}_3}}+\frac{1}{6}\underset{n_1=n_3}{\sum _{(n_1,n_2,n_3)\in {\mathcal{C}}_3}}+\frac{1}{6}\underset{n_2=n_3}{\sum _{(n_1,n_2,n_3)\in {\mathcal{C}}_3}}\right)$$

$$+\frac{1}{3}\underset{n_1=n_2=n_3}{\sum _{(n_1,n_2,n_3)\in {\mathcal{C}}_3}}]\frac{1}{(n_1{n}_2{n}_3{)}^s}$$

$$=[\frac{1}{6}\left(\sum _{\text{all ni unequal}}+\sum _{n_1=n_2\ne n_3}+\sum _{n_1=n_3\ne n_2}+\sum _{n_1\ne n_2=n_3}+\sum _{n_1=n_2=n_3}\right)$$

$$+\frac{1}{6}\left(\sum _{n_1=n_2\ne n_3}+\sum _{n_1=n_2=n_3}\right)+\frac{1}{6}\left(\sum _{n_1=n_3\ne n_2}+\sum _{n_1=n_2=n_3}\right)$$

$$+\frac{1}{6}\left(\sum _{n_1\ne n_2=n_3}+\sum _{n_1=n_2=n_3}\right)+\frac{1}{3}\sum _{n_1=n_2=n_3}]\frac{1}{(n_1{n}_2{n}_3{)}^s}$$

$$=\left[\frac{1}{6}\left(\sum _{\text{all ni unequal}}\right)+\frac{1}{3}\left(\sum _{n_1=n_2\ne n_3}+\sum _{n_1=n_3\ne n_2}+\sum _{n_1\ne n_2=n_3}\right)+1\sum _{n_1=n_2=n_3}\right]\frac{1}{(n_1{n}_2{n}_3{)}^s}$$

$$=\left[\frac{1}{6}\sum _{\text{all ni unequal}}+\frac{1}{3}\underset{n_1\lessgtr n_2=n_3}{\underset{n_1=n_3\lessgtr n_2,\mathrm{or}}{\sum _{n_1=n_2\lessgtr n_3,\mathrm{or}}}}+\sum _{n_1=n_2=n_3}\right]\frac{1}{(n_1{n}_2{n}_3{)}^s}$$

$$=\left[\sum _{n_1>n_2>n_3\ge 1}+\left(\sum _{n_1>n_2=n_3\ge 1}+\sum _{n_1=n_2>n_3\ge 1}\right)+\sum _{n_1=n_2=n_3}\right]\frac{1}{(n_1{n}_2{n}_3{)}^s}$$

$$=\sum _{\begin{array}{c}\pi \in \mathcal{P}\\ \ell (\pi )=3\end{array}}\frac{1}{N(\pi {)}^s}={\zeta }_{\mathcal{P}}(\{s{\}}^3),$$

$$\prod _{j=1}^k\frac{1}{1-q^j}=\sum _{\lambda ⊢k}\frac{1}{N(\lambda )m_1!m_2!m_3!\cdots m_k!(1-q{)}^{m_1}(1-q^2{)}^{m_2}(1-q^3{)}^{m_3}\cdots (1-q^k{)}^{m_k}}.$$

$$\frac{q^k}{{\prod }_{j=1}^k(1-q^j)}=\sum _{\lambda ⊢k}\frac{q^{m_1}{q}^{2m_2}{q}^{3m_3}\cdots q^{k{m}_k}}{N(\lambda )m_1!m_2!m_3!\cdots m_k!(1-q{)}^{m_1}(1-q^2{)}^{m_2}(1-q^3{)}^{m_3}\cdots (1-q^k{)}^{m_k}}.$$

$$\frac{q^k}{{\prod }_{j=1}^k(1-q^j)}=\sum _{\ell (\lambda )=k}{q}^{|\lambda |}⟷{\zeta }_{\mathcal{P}}({\left\{s\right\}}^k)=\sum _{\ell (\lambda )=k}N(\lambda {)}^{-s}.$$

$$q^{{\lambda }_1}{q}^{{\lambda }_2}{q}^{{\lambda }_3}\cdots q^{{\lambda }_r}=q^{|\lambda |}⟷{\lambda }_1^{-s}{\lambda }_2^{-s}{\lambda }_3^{-s}\cdots {\lambda }_r^{-s}=N{\left(\lambda \right)}^{-s}.$$

$$\frac{q^j}{1-q^j}=\sum _{n=1}^{\infty }{q}^{|(n{)}^j|}⟷\zeta (js)=\sum _{n=1}^{\infty }N{\left((n{)}^j\right)}^{-s}.$$

Full text

Dedicated to Bruce Berndt on the occasion of his $80$th birthday

1. Introduction: partition zeta functions

Here we study an interesting class of objects dwelling at the intersection of partition theory and the theory of L-functions. Let $\mathcal{P}$ denote the set of integer partitions (see e.g. [1, 2]), with $\mathcal{P}_{S}$ being partitions whose parts all belong to a subset $S\subseteq\mathbb{N}$ of natural numbers. Let $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},...,\lambda_{r})$ denote a generic partition, $\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{r}\geq 1$, with $\emptyset$ the empty partition. Let $|\lambda|$ denote the sum of parts or size of $\lambda$, with $|\emptyset|:=0$. Let $N(\lambda)$ denote the product of the parts or norm of the partition, with $N(\emptyset):=1$. Let $\ell(\lambda):=r$ denote the number of parts or length of a partition, with $\ell(\emptyset):=0$.

Then in analogy to the Riemann zeta function $\zeta(s):=\sum_{n=1}^{\infty}{n^{-s}}$, convergent for $\operatorname{Re}(s)>1$, we give the definition of a partition zeta function as studied in [5, 7, 8].

Definition 1.1**.**

For a proper subset $\mathcal{P}^{\prime}\subset\mathcal{P}$ and value $s\in\mathbb{C}$ for which the series converges, we define a partition zeta function to be the following sum over partitions in $\mathcal{P}^{\prime}$:

[TABLE]

We note that $\zeta_{\mathcal{P}}(s)$ itself diverges111And similarly if the maximum number of $1$’s appearing in partitions in $\mathcal{P}^{\prime}$ is unrestricted: partitions of the shape $\lambda=(1,1,1,...,1)$ contribute infinitely many $1$’s to the sum. If $\mathcal{P}^{\prime}$ is of the class $\mathcal{P}_{S}$ of partitions into elements from some subset $S\subsetneq\mathbb{N}$ where $1\not\in S$, then the associated zeta function has an Euler product as well:

[TABLE]

and standard techniques working with products give other variants.

Of course, in this setting the Riemann zeta function $\zeta(s)$ represents the case $\zeta_{\mathcal{P}_{\mathbb{P}}}(s)$ (sum over partitions into prime parts). These combinatorial zeta functions (as well as related partition Dirichlet series) form a highly general class of objects, yet they share many structural laws, which specialize to well-known classical zeta function (and Dirichlet series) identities, as well as more exotic non-classical cases such as the following, proved in [7], for partitions into even parts, distinct parts, and parts not equal to 1, respectively:

[TABLE]

Since choice of subset $\mathcal{P}^{\prime}\subsetneq\mathcal{P}$ as well as domain of $s\in\mathbb{C}$ determine a given partition zeta function, each pair of choices yields distinctive yet disparate evaluations and analytic properties.

Then identifying interesting non-classical cases seems like something of a needle-in-a-haystack type of undertaking. One wonders: are there non-classical families of partition zeta functions having such nice properties that they truly parallel, for instance, Euler’s evaluations of $\zeta(s)$ at positive even arguments:

[TABLE]

2. A nice family of partition zeta functions

A particularly nice class of zeta functions is defined in [7], that generalizes $\zeta(s)$ in a more fruitful direction than just writing $\zeta(s)=\zeta_{\mathcal{P}_{\mathbb{P}}}(s)$.

Definition 2.1**.**

For $\operatorname{Re}(s)>1$, we define the family of zeta sums taken over all partitions of fixed length $k\geq 0$:

[TABLE]

with $\zeta_{\mathcal{P}}(\{s\}^{0}):=N(\emptyset)^{-s}=1$.

The $k=1$ case is $\zeta(s)$. For $|z|<1$ we have the obvious generating function:

[TABLE]

The first author proves in [7] that for argument $s=2$ and $k\geq 1$, these partition zeta functions are actually rational multiples of Euler’s zeta values (and of $\pi^{2k}$).

Theorem** (Schneider).**

For $k\geq 1$ we have

[TABLE]

We note that setting $k=0$ in Proposition Theorem suggests formally that $\zeta(0)=\frac{2^{-2}}{2^{-1}-1}\zeta_{\mathcal{P}}(\{2\}^{0})=-1/2$, which is the correct value for $\zeta(0)$ obtained through analytic continuation. In [5], Ono, Rolen and the first author prove other facts about partition zeta functions, including special cases yielding analytic continuation (restricted, however, to the right half of the complex plane), and a farther-reaching follow-up to the preceding theorem.

Theorem** (Ono–Rolen–Schneider).**

For $m\geq 1,k\geq 1,$ we have

[TABLE]

So these zeta sums over partitions of fixed length do indeed form a family like Euler’s zeta values; but zeta values are only the beginning of the story of the zeta function, as Riemann displayed in a brilliant sequel to Euler’s work. Then it is natural to ask about analytic properties of the general case $\zeta_{\mathcal{P}}(\{s\}^{k}),\operatorname{Re}(s)>1$. Here we prove a closed formula for the partition zeta function for each $k\geq 0$, that speaks to this question.

Let “$\lambda\vdash n$” mean that $\lambda$ is a partition of size $n\geq 0$, and let $m_{j}=m_{j}(\lambda)\geq 0$ denote the multiplicity of $j\geq 1$ as a part of partition $\lambda$.

Theorem 2.2**.**

For $\operatorname{Re}(s)>1,k\geq 0$, we have

[TABLE]

where the sum on the right-hand side is taken over the partitions of size $k\geq 0$.

To prove the theorem, we need the Maclaurin expansion of the natural logarithm for $|x|<1$:

[TABLE]

as well as a partition version of Faà di Bruno’s formula. For completeness, we give a quick proof of the classical identity below.

Faà di Bruno’s formula**.**

Take $\operatorname{exp}(t):=e^{t}$ and $a_{j},x\in\mathbb{C}$ such that $\sum_{j=1}^{\infty}a_{j}x^{j}$ converges. Then we have

[TABLE]

Proof of Faà di Bruno’s formula.

We begin with the classical multinomial theorem, written as a sum over length-$n$ partitions $\lambda$ in the set $\mathcal{P}_{[k]}\subset\mathcal{P}$ whose parts are all $\leq k$:

[TABLE]

If we let $k$ tend to infinity, assuming the infinite sum $a_{1}+a_{2}+a_{3}+\cdots$ converges, the series on the right becomes a sum over all partitions of length $n$. Then dividing both sides of (2.3) by $n!$ and summing over $n\geq 0$, the left-hand side yields the Maclaurin series expansion for $\operatorname{exp}(a_{1}+a_{2}+a_{3}+\cdots)$, and the right side can be rewritten as a sum over all partitions:

[TABLE]

To complete the proof, make the substitution $a_{k}\mapsto a_{k}x^{k}$ in (2.4). ∎

Proof of Theorem 2.2.

For $|z|<1,\operatorname{Re}(s)>1$, by (2.2) we rewrite the product side of generating function (2.1) as:

[TABLE]

Now setting $x=z$ and $a_{j}=\zeta(js)/j$ in Lemma Faà di Bruno’s formula and noting $N(\lambda)=\prod_{j\geq 1}j^{m_{j}}$ for each partition $\lambda$, comparing the coëfficient of $z^{k}$ with the right-hand side of (2.1) gives the theorem. ∎

Theorem 2.2 yields much information about the analytic properties of $\zeta_{\mathcal{P}}(\{s\}^{k})$.

Corollary 2.3**.**

The partition zeta function $\zeta_{\mathcal{P}}(\{s\}^{k})$:

1. i.

inherits analytic continuation from $\zeta(s)$, to the entire complex plane minus poles; 2. ii.

has poles at $s=1,\ 1/2,\ 1/3,\ 1/4,...,1/k$ with the order of $s=1/j$ being $\left\lfloor k/j\right\rfloor$; 3. iii.

has trivial roots at $s=-2,-4,-6,-8,$ etc.; 4. iv.

does not have roots at the nontrivial roots of $\zeta(s)$.

Proof.

All four items of the corollary follow immediately from well-known analytic properties of the Riemann zeta function. That $\zeta_{\mathcal{P}}(\{s\}^{k})$ inherits analytic continuation from the finite combination of $\zeta(js)$ factors is obvious, and the pole at $s=1/j$ comes from the pole of $\zeta(js)$ at $js=1$ with order of the pole coming from the partition of $k$ with the maximum possible number $\left\lfloor k/j\right\rfloor$ of $j$’s. For $s\in-2\mathbb{N}$, all the $\zeta(js)$ vanish trivially, giving trivial roots of the partition zeta function; but on the assumption no single value $s$ can serve as a nontrivial root for all the $\zeta(js),1\leq j\leq k,$ simultaneously, the partition zeta function does not entirely vanish at any nontrivial root of $\zeta(s)$. ∎

3. Combinatorial approach

By contrast with the analytic proof of the previous section, in this section we give a combinatorial proof of Theorem 2.2. We take an almost identical approach to the second author’s work [9] generating partitions of fixed length and allowing the symmetric group to act on these partitions to create integer compositions.

Let $\mathcal{C}_{k}$ denote the set of $k$-tuples of positive integers $c=(c_{1},c_{2},...,c_{k})$. These are integer compositions of length $k$ (or “$k$-compositions”). As with partitions, the $c_{j}$ in $c=(c_{1},\dots,c_{k})$ are the parts of $c$, and the sum of parts $|c|$ is the size of $c$.

We proceed by example. First we recall that $\zeta_{\mathcal{P}}(\left\{s\right\}^{0})=1$ and the $k=1$ case is simply

[TABLE]

The $k=2$ case of Theorem 2.2 may be proved combinatorially as follows:

[TABLE]

The details for the case $k=3$ are as follows:

[TABLE]

where “$a\lessgtr b$” means $a\neq b$, but we mean to emphasize that only two of the six cases where exactly two of the $n_{i}$ are equal, namely “$n_{1}>n_{2}=n_{3}$” and “$n_{1}=n_{2}>n_{3}$,” yield a partition.

As the length $k$ increases, analogous combinatorial principles of splitting up, sorting compositions and regrouping are at play in the indices of summation. For the corresponding proof of the general $k$ case, the reader is referred to [9]; while they are too lengthy to conveniently reproduce here, the exact steps of the proof of MacMahon’s partial fraction formula given in [9] also prove Theorem 2.2 under certain substitutions, which we detail in the next section.

4. Correspondence with MacMahon partial fractions

The combinatorial identities in Section 3 mimic the second author’s proof [9] of MacMahon’s partial fraction decomposition of the generating function for partitions of length $\leq k$ (see [3, 4]). Let us recall MacMahon’s result.

MacMahon’s decomposition**.**

For $|q|<1$, we have

[TABLE]

Multiplying both sides of (4.1) by $q^{k}$ and noting for every partition of $k$ that $k=m_{1}+2m_{2}+3m_{3}+\cdots+km_{k}$, gives the generating function for partitions of length exactly $k$:

[TABLE]

We point out that there is a simple bijection between partitions generated in (4.2) and those in (4.1): for a partition $\lambda=(\lambda_{1},\lambda_{2},\dots,\lambda_{j})$ of length $j\leq k$, identify $\lambda$ with the $k$-tuple $(\lambda_{1},\lambda_{2},\dots,\lambda_{j},0,0,\dots,0)$ and then add $1$ to each component of the $k$-tuple to obtain a partition of length exactly $k$.

Then comparing equation (4.2) to Theorem 2.2, one sees an apparent correspondence between $q$-series generating functions and the respective zeta function components in Theorem 2.2.222Similar correspondences between $q$-series and zeta functions have been studied by the first author and his collaborators (see [5, 6, 7, 8]). We summarize these observations in the table below; corresponding entries evidently encode the same partition-theoretic information:

[TABLE]

The analogy between the left-hand sides of (4.2) and Theorem 2.2 is clear when we compare the following summation representations:

[TABLE]

There is a one-to-one correspondence between the terms in each summation, i.e., between the partitions being generated. We note that multiplication of terms of either shape $q^{n}$ or $n^{-s}$ generates partitions in exactly the same way, viz. for partition $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3},...,\lambda_{r})$:

[TABLE]

Therefore, taken in finite combinations, geometric series and zeta functions behave identically as partition generating functions333For infinite combinations, a different analogy holds, between product generating functions like $\prod_{j=1}^{\infty}(1-q^{j})^{-1}$ and corresponding Euler products (see [7]).. Whereas the size $|\lambda|$ is the partition-encoding statistic in the former scheme, the norm $N(\lambda)$ encodes partitions in the latter.

Moreover, the summand $q^{jn}$ in the geometric series $\sum_{n=1}^{\infty}q^{jn}$ and the respective term $n^{-js}$ of $\zeta(js)$ can both be viewed as encoding the partition $(n)^{j}:=(n,n,...,n)$ consisting of $j$ repetitions of the part $n$, viz.

[TABLE]

The $n$th terms of the summations in (4.5) are in one-to-one correspondence.

Now, since the multiplicities $m_{1},m_{2},...,m_{k}$ associated to any length-$k$ partition must add up to $k$, these nonzero multiplicities themselves represent a permutation of some partition of size $k$. This observation brings compositions into the picture, and also provides a link between partitions of length $k$ and those of size $k$.

To prove MacMahon’s partial fraction formula in [9], roughly speaking, one replaces the term $q^{j}$ with a multivariate product $x_{1}x_{2}x_{3}\cdots x_{j},\ |x_{i}|<1$, then permutes the $x_{i}$’s. More exactly, the geometric factors in the right-hand summands of (4.1) will be rewritten

[TABLE]

One then permutes the $x_{i}$’s of nonempty geometric factors, effectively giving rise to compositions, while keeping track of the permutations in the indices of summation as in the examples in Section 3, using properties of the symmetric group to enumerate multiply-counted terms. In the end one sets each dummy variable $x_{i}=q$ to produce (4.1).

The counting arguments of [9] are unaltered in the case of partitions of length exactly $k$. One can rewrite the geometric aspect of each right-hand summand of (4.2) as

[TABLE]

and by precisely the same steps taken in [9], equation (4.2) is proved. Going a step further in this direction: as $|x_{i}|<1$, one can rewrite the $j$th geometric factor of (4.7) above in series form:

[TABLE]

noting $i_{m+1}=i_{m}+1$ in the indices, then in the sum on the right side of (4.8) make the change

[TABLE]

with the $x_{i}$’s now needing to satisfy $\operatorname{Re}(x_{i_{1}}+x_{i_{2}}+\cdots+x_{i_{j}})>1$ for convergence of the series. This mapping produces a one-to-one term-wise correspondence:

[TABLE]

Moreover, the terms of both these series are in one-to-one correspondence with the summands of the $j$th geometric factor of (4.6) when it is expanded as a series:

[TABLE]

and the symmetric group acts on the $x_{i}$’s identically in finite combinations of series of any one of these types.

Then one can replace the $j$th geometric factor of (4.6), $j=1,2,3,...,k$, with the right-hand multivariate zeta series in (4.10), permute the $x_{i}$’s and enumerate permutations following exactly the steps in [9]; finally, one sets each $x_{i}=s$ such that $x_{i_{1}}+x_{i_{2}}+\cdots+x_{i_{j}}=js$, to arrive at Theorem 2.2.