{
  "id": "2109.00609",
  "version": "v1",
  "published": "2021-09-01T20:49:11.000Z",
  "updated": "2021-09-01T20:49:11.000Z",
  "title": "On a Partition Identity of Lehmer",
  "authors": [
    "Cristina Ballantine",
    "Hannah E. Burson",
    "Amanda Folsom",
    "Chi-Yun Hsu",
    "Isabella Negrini",
    "Boya Wen"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called \"Beck-type\" companions to other identities. In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-01T20:49:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "05A19",
      "11P83"
    ],
    "keywords": [
      "partition identity",
      "odd parts",
      "related beck-type companion identities",
      "becks original conjecture",
      "eulers identity equates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}