Symmetric polynomials associated with numerical semigroups

TL;DR

This paper introduces a new class of symmetric polynomials linked to numerical semigroups, explores their properties, and suggests a conjectural connection to partition functions, enriching the mathematical understanding of these structures.

Contribution

It defines and analyzes a novel type of symmetric polynomials associated with numerical semigroups and proposes a conjecture relating them to partition functions.

Findings

Established basic properties of the new polynomials

Derived their representation via power sums

Observed a visual similarity to partition function components

Abstract

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums E_k=\sum_{j=1}^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x_1,...,x_m)/\chi_m, where \chi_m=\prod_{j=1}^m x_j, and a polynomial part of a partition function W(s,{d_1,...,d_m}), which gives a number of partitions of s\ge 0 into m positive integers d_j, and put forward a conjecture about their relationship.