Cyclic Sieving of Multisets with Bounded Multiplicity and the Frobenius Coin Problem
Drew Armstrong
TL;DR
The paper investigates cyclic sieving for multisets with bounded multiplicity by studying principal specializations of the $b$-bounded symmetric polynomials $h_k^{(b)}$ and their $q$-analogs. It derives generating-function formulas for $\genfrac{[}{]}{0pt}{}{n}{k}^{(b)}_\omega$ at primitive $d$-th roots of unity across three divisibility regimes, establishing CSP when $d|n$ or $d|(n-1)$ under $\gcd(b,d)=1$, and revealing a Frobenius coin problem connection in the case $d|(n+1)$. The work further connects these evaluations to combinatorial interpretations for bounded multisets, and it develops a detailed Sylvester coin framework (including a double-abacus) to describe the coefficients in the $d|(n+1)$ case, alongside comprehensive expansions of $h_k^{(b)}$ in various symmetric-function bases. Altogether, the results unify and extend CSP for bounded multiplicities, illuminate number-theoretic links via the Frobenius problem, and situate $b$-bounded symmetric polynomials within the broader theory of symmetric functions.
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 $ω$ be a primitive $d$th root of unity. If $d|n$ or $d|(n-1)$ then we have $f(1,\ldots,ω^{n-1})\in\mathbb{Z}$. If $d|n$ then of course we have $f(ω,\ldots,ω^n)=f(1,\ldots,ω^{n-1})\in\mathbb{Z}$, but when $d|(n+1)$ we also have $f(ω,\ldots,ω^n)\in\mathbb{Z}$. 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 $h_k^{(b)}=(1,ω,\ldots,ω^{n-1})$ 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 $h_k^{(b)}(ω,ω^2,\ldots,ω^n)$ 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.