标题： 多重集的循环筛选与有界重数及Frobenius硬币问题 显示英文标题

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

摘要： 标题中的两个主题通过单位根处对称多项式的特殊化相关联。 设 $f(z_1,\ldots,z_n)\in\mathbb{Z}[z_1,\ldots,z_n]$是一个具有整数系数的对称多项式，令$\omega$是一个原根$d$次单位根。 如果 $d|n$或$d|(n-1)$则我们有$f(1,\ldots,\omega^{n-1})\in\mathbb{Z}$。 如果$d|n$，那么当然我们有$f(\omega,\ldots,\omega^n)=f(1,\ldots,\omega^{n-1})\in\mathbb{Z}$，但当$d|(n+1)$时我们也得到$f(\omega,\ldots,\omega^n)\in\mathbb{Z}$。 我们研究这三个整数族在$f=h_k^{(b)}$的情况下，其中$h_k^{(b)}$是生成函数$\prod_{i=1}^n (1+z_it+\cdots+(z_it)^{b-1})$中$t^k$的系数。 这些多项式之前已被几位作者考虑过。 它们在基本对称多项式（$b$=2）和完全齐次对称多项式（$b\to\infty$）之间进行插值。 当$\gcd(b,d)=1$与$d|n$或$d|(n-1)$时，我们发现整数$h_k^{(b)}=(1,\omega,\ldots,\omega^{n-1})$与多重集的循环筛分有关，其重数上限为$b$，这推广了集合（$b=2$）和多重集（$b\to \infty$）的著名循环筛分结果。 当$\gcd(b,d)=1$和$d|(n+1)$时，我们发现整数$h_k^{(b)}(\omega,\omega^2,\ldots,\omega^n)$与两个硬币的Frobenius邮票问题有关。情况$\gcd(b,d)\neq 1$更为复杂。在论文结尾，我们将这些结果与$h_k^{(b)}$在对称多项式环的各种基中的展开相结合。 显示英文摘要

摘要： 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.