Cyclic Sieving of Multisets with Bounded Multiplicity and the Frobenius Coin Problem

TL;DR

This paper explores the connections between symmetric polynomials evaluated at roots of unity, cyclic sieving phenomena for multisets with bounded multiplicity, and the Frobenius coin problem, revealing new algebraic and combinatorial insights.

Contribution

It introduces new relationships between symmetric polynomial evaluations, cyclic sieving for bounded multisets, and the Frobenius coin problem, extending previous results to more general settings.

Findings

Integer evaluations relate to cyclic sieving of bounded multisets.

Connections established between symmetric polynomial evaluations and the Frobenius coin problem.

Generalization of cyclic sieving results from sets and multisets to bounded multisets.

Abstract

The two subjects in the title are related via the specialization of symmetric polynomials at roots of unity. Let $f(z_1,\ldots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n]$ be a symmetric polynomial with integer coefficients and let $\omega$ be a primitive $d$th root of unity. If $d|n$ or $d|(n-1)$ then we have $f(1,\ldots,\omega^{n-1})\in\mathbb{Z}$. If $d|n$ then of course we have $f(\omega,\ldots,\omega^n)=f(1,\ldots,\omega^{n-1})\in\mathbb{Z}$, but when $d|(n+1)$ we also have $f(\omega,\ldots,\omega^n)\in\mathbb{Z}$. We investigate these three families of integers in the case $f=h_k^{(b)}$, where $h_k^{(b)}$ is the coefficient of $t^k$ in the generating function $\prod_{i=1}^n (1+z_it+\cdots+(z_it)^{b-1})$. These polynomials were previously considered by several authors. They interpolate between the elementary symmetric polynomials ($b$=2) and the complete homogeneous symmetric polynomials ($b\to\infty$). When $\gcd(b,d)=1$ with $d|n$ or $d|(n-1)$ we find that the integers $h_k^{(b)}=(1,\omega,\ldots,\omega^{n-1})$ are related to cyclic sieving of multisets with multiplicities bounded above by $b$, generalizing the well know cyclic sieving results for sets ($b=2$) and multisets ($b\to \infty$). When $\gcd(b,d)=1$ and $d|(n+1)$ we find that the integers $h_k^{(b)}(\omega,\omega^2,\ldots,\omega^n)$ are related to the Frobenius coin problem with two coins. The case $\gcd(b,d)\neq 1$ is more complicated. At the end of the paper we combine these results with the expansion of $h_k^{(b)}$ in various bases of the ring of symmetric polynomials.