{
  "id": "1610.08726",
  "version": "v1",
  "published": "2016-10-27T11:58:45.000Z",
  "updated": "2016-10-27T11:58:45.000Z",
  "title": "Wilf's conjecture for numerical semigroups",
  "authors": [
    "Mariam Dhayni"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $S\\subseteq \\N$ be a numerical semigroup with multiplicity $m$, embedding dimension $\\nu$ and conductor $c=f+1=qm-\\rho$ for some $q,\\rho\\in\\N$ with $\\rho\\textless{}m$. Let Ap$(S,m) = \\{w\\_0\\textless{}w\\_1 \\textless{} \\ldots \\textless{}w\\_{m-1}\\}$ be the Ap\\'ery set of $S$. The aim of this paper is to prove Wilf's Conjecture in some special cases. First, we prove that if $w\\_{m-1}\\geq w\\_1+w\\_{\\alpha}$ and $(2+\\frac{\\alpha-3}{q})\\nu\\geq m$ for some $1\\textless{}\\alpha\\textless{}m-1$, then $S$ satisfies Wilf's Conjecture. Then, we prove the conjecture in the following cases: $(2+\\frac{1}{q})\\nu\\geq m$, $m-\\nu\\leq 5$ and $m=9$. Finally, the conjecture is proved if $w\\_{m-1}\\geq w\\_{\\alpha-1}+w\\_{\\alpha}$ and $(\\frac{\\alpha+3}{3})\\nu\\geq m$ for some $1\\textless{}\\alpha\\textless{}m-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-27T11:58:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical semigroup",
      "satisfies wilfs conjecture",
      "apery set",
      "special cases",
      "embedding dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}