{
  "id": "2502.00378",
  "version": "v1",
  "published": "2025-02-01T09:32:32.000Z",
  "updated": "2025-02-01T09:32:32.000Z",
  "title": "Cyclic Sieving of Multisets with Bounded Multiplicity and the Frobenius Coin Problem",
  "authors": [
    "Drew Armstrong"
  ],
  "comment": "36 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-01T09:32:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "frobenius coin problem",
      "cyclic sieving",
      "bounded multiplicity",
      "elementary symmetric polynomials",
      "complete homogeneous symmetric polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}