Symmetric polynomials associated with numerical semigroups

2020/10/07 by Fel, Leonid G. · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #Primary - 20M14 #Secondary - 11P81

paper · doi:10.48550/arxiv.2010.03363

Abstract

We study a new kind of symmetric polynomials P_n(x₁,...,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=∑_j=1^m x_j^k. We observe a visual similarity between normalized polynomials P_n(x₁,...,x_m)/χ_m, where χ_m=∏_j=1^m x_j, and a polynomial part of a partition function W(s,d₁,...,d_m), which gives a number of partitions of s≥ 0 into m positive integers d_j, and put forward a conjecture about their relationship.

Cited by

• A New Class of Relations for Homogeneous Symmetric Polynomials
• Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

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