{
  "id": "2011.08220",
  "version": "v1",
  "published": "2020-11-16T19:06:59.000Z",
  "updated": "2020-11-16T19:06:59.000Z",
  "title": "Beck-type identities: new combinatorial proofs and a theorem for parts congruent to $t$ mod $r$",
  "authors": [
    "Cristina Ballantine",
    "Amanda Welch"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\mathcal O_r(n)$ be the set of $r$-regular partitions of $n$, $\\mathcal D_r(n)$ the set of partitions of $n$ with parts repeated at most $r-1$ times, $\\mathcal O_{1,r}(n)$ the set of partitions with exactly one part (possibly repeated) divisible by $r$, and let $\\mathcal D_{1,r}(n)$ be the set of partitions in which exactly one part appears at least $r$ times. If $E_{r, t}(n)$ is the excess in the number of parts congruent to $t \\pmod r$ in all partitions in $\\mathcal O_r(n)$ over the number of different parts appearing at least $t$ times in all partitions in $\\mathcal D_r(n)$, then $E_{r, t}(n) = |\\mathcal O_{1,r}(n)| = |\\mathcal D_{1,r}(n)|$. We prove this analytically and combinatorially using a bijection due to Xiong and Keith. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in $\\mathcal{O}_r(n)$ over the number of parts in all partitions in $\\mathcal{D}_r(n)$ equals $(r - 1)|\\mathcal{O}_{1,r}(n)|$ and also $(r - 1)|\\mathcal{D}_{1,r}(n)|$. Our work provides a new combinatorial proof of this result that does not use Glaisher's bijection. We also give a new combinatorial proof based of the Xiong-Keith bijection for a second Beck-Type identity that has been proved previously using Glaisher's bijection.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-16T19:06:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P81",
      "11P83"
    ],
    "keywords": [
      "combinatorial proof",
      "parts congruent",
      "glaishers bijection",
      "first beck-type identity",
      "second beck-type identity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}