{
  "id": "2406.00139",
  "version": "v1",
  "published": "2024-05-31T19:03:44.000Z",
  "updated": "2024-05-31T19:03:44.000Z",
  "title": "Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity",
  "authors": [
    "Cristina Ballantine",
    "Amanda Welch"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We consider the number of various partitions of $n$ with parts separated by parity and prove combinatorially several inequalities between these numbers. For example, we show that for $n\\geq 5$ we have $p_{od}^{eu}(n)<p_{ed}^{ou}(n)$, where $p_{od}^{eu}(n)$ is the number of partitions of $n$ with odd parts distinct and even parts unrestricted and all odd parts less than all even parts and $p_{ed}^{ou}(n)$ is the number of partitions of $n$ with even parts distinct and odd parts unrestricted and all even parts less than all odd parts. We also prove a conjectural inequality of Fu and Tang involving partitions with parts separated by parity with restrictions on the multiplicity of parts.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-31T19:03:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "05A20",
      "11P81"
    ],
    "keywords": [
      "partitions",
      "combinatorial proofs",
      "odd parts distinct",
      "conjectural inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}