Reciprocals of Subsum Polynomials

TL;DR

This paper investigates the reciprocals of subsum polynomials derived from partitions, exploring their arithmetic properties and connections to combinatorial objects.

Contribution

It introduces the concept of subsum polynomials for partitions and analyzes the properties of their reciprocals across all partitions of a given integer.

Findings

Proves arithmetic properties of reciprocals of subsum polynomials.

Establishes connections between these reciprocals and other combinatorial objects.

Abstract

We introduce the subsum polynomial of a partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ defined by $\mathrm{sp}(\lambda, x)=\prod_{i=1}^k(1+x^{\lambda_i})$. We study the sum of reciprocals of $\mathrm{sp}(\lambda, x)$ over all partitions of $n$. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.