{
  "id": "2008.08110",
  "version": "v1",
  "published": "2020-08-18T18:14:36.000Z",
  "updated": "2020-08-18T18:14:36.000Z",
  "title": "Numerical Semigroups of small and large type",
  "authors": [
    "Deepesh Singhal"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup $\\frac{g}{F+1-g}\\leq t\\leq 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterises them. We give an explicit characterisation of numerical semigroups with $t=\\frac{g}{F+1-g}$. We show that for a fixed $\\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\\alpha$ is eventually constant for large $F$. Also the number of numerical semigroups with genus $g$ and type $g-\\alpha$ is also eventually constant for large $g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-18T18:14:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical semigroup",
      "large type",
      "frobenius number",
      "eventually constant",
      "finite complement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}