{
  "id": "2410.16985",
  "version": "v1",
  "published": "2024-10-22T13:06:49.000Z",
  "updated": "2024-10-22T13:06:49.000Z",
  "title": "Sequences of odd length in strict partitions II: the $2$-measure and refinements of Euler's theorem",
  "authors": [
    "Shishuo Fu",
    "Haijun Li"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The number of sequences of odd length in strict partitions (denoted as $\\mathrm{sol}$), which plays a pivotal role in the first paper of this series, is investigated in different contexts, both new and old. Namely, we first note a direct link between $\\mathrm{sol}$ and the $2$-measure of strict partitions when the partition length is given. This notion of $2$-measure of a partition was introduced quite recently by Andrews, Bhattacharjee, and Dastidar. We establish a $q$-series identity in three ways, one of them features a Franklin-type involuion. Secondly, still with this new partition statistic $\\mathrm{sol}$ in mind, we revisit Euler's partition theorem through the lens of Sylvester-Bessenrodt. Two new bivariate refinements of Euler's theorem are established, which involve notions such as MacMahon's 2-modular Ferrers diagram, the Durfee side of partitions, and certain alternating index of partitions that we believe is introduced here for the first time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-22T13:06:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P84",
      "05A17",
      "05A15"
    ],
    "keywords": [
      "strict partitions",
      "eulers theorem",
      "odd length",
      "refinements",
      "revisit eulers partition theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}