Partitions and elementary symmetric polynomials -- an experimental approach

TL;DR

This paper explores elementary symmetric polynomials evaluated at partitions, establishing analogs of classical partition formulas, proving congruences, and proposing conjectures on combinatorial properties and log-concavity of related functions.

Contribution

It introduces new formulas and congruences for sums of elementary symmetric polynomials over partitions, and formulates conjectures on their combinatorial and algebraic properties.

Findings

Proved analogs of classical partition formulas for elementary symmetric polynomials.

Established congruences for sums over partitions into four parts.

Conjectured inequalities and log-concavity properties related to these functions.

Abstract

Given a partition $\lambda$, we write $e_j(\lambda)$ for the $j^{\textrm{th}}$ elementary symmetric polynomial $e_j$ evaluated at the parts of $\lambda$ and $e_jp_A(n)$ for the sum of $e_j(\lambda)$ as $\lambda$ ranges over the set of partitions of $n$ with parts in $A$. For $e_jp_A(n)$, we prove analogs of the classical formula for the partition function, $p(n)=1/n \sum_{k=0}^{n-1}\sigma_1(n-k)p(k)$, where $\sigma_1$ is the sum of divisors function. We prove several congruences for $e_2p_4(n)$, the sum of $e_2$ over the set of partitions of $n$ into four parts. Define the function $\textrm{pre}_j(\lambda)$ to be the multiset of monomials in $e_j(\lambda)$, which is itself a partition. If $\mathcal A$ is a set of partitions, we define $\textrm{pre}_j(\mathcal A)$ to be the set of partitions $\textrm{pre}_j(\lambda)$ as $\lambda$ ranges over $\mathcal A$. If $\mathcal P(n)$ is the set of all partitions of $n$, we conjecture that the number of odd partitions in $\textrm{pre}_2(\mathcal P(n))$ is at least the number of distinct partitions. We prove some results about $\textrm{pre}_2(\mathcal B(n))$, where $\mathcal B(n)$ is the set of binary partitions of $n$. We conclude with conjectures on the log-concavity of functions related to $e_jp(n)$, the sum of $e_j(\lambda)$ for all $\lambda\in \mathcal P(n)$.