{
  "id": "1710.05960",
  "version": "v1",
  "published": "2017-10-16T18:50:48.000Z",
  "updated": "2017-10-16T18:50:48.000Z",
  "title": "Bisected theta series, least $r$-gaps in partitions, and polygonal numbers",
  "authors": [
    "Cristina Ballantine",
    "Mircea Merca"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "The least $r$-gap, $g_r(\\lambda)$, of a partition $\\lambda$ is the smallest part of $\\lambda$ appearing less than $r$ times. In this article we introduce two new partition functions involving least $r$-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler's partition function $p(n)$, polygonal numbers, and the new partition functions. To prove the results we use an interplay of combinatorial and $q$-series methods. We also give a combinatorial interpretation for $$\\sum_{n=0}^\\infty (\\pm 1)^{k(k+1)/2} p(n-r\\cdot k(k+1)/2).$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-16T18:50:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "bisected theta series",
      "polygonal numbers",
      "identities relating eulers partition function",
      "series methods",
      "combinatorial interpretation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}