{
  "id": "2110.15837",
  "version": "v2",
  "published": "2021-10-29T15:00:15.000Z",
  "updated": "2022-01-18T15:51:45.000Z",
  "title": "Self-conjugate $t$-core partitions and applications",
  "authors": [
    "Madeline Locus Dawsey",
    "Benjamin Sharp"
  ],
  "comment": "15 pages, accepted for publication in Australasian Journal of Combinatorics",
  "journal": "Australas. J. Combin. 82(2) (2022), 212--227",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Partition theory abounds with bijections between different types of partitions. One of the most famous partition bijections maps each self-conjugate partition of a positive integer $n$ to a partition of $n$ into distinct odd parts, and vice versa. Here we prove new necessary and sufficient conditions for a self-conjugate partition to be $t$-core, in terms of only the parts of the corresponding partition into distinct odd parts, by proving a new hook length formula. Corollaries of these results include new applications of $t$-core self-conjugate partitions to subsets of the natural numbers, due to the recent investigation of a new partition statistic called the supernorm by the first author, Just, and Schneider, as well as many results on $t$-cores by Bringmann, Kane, Males, Ono, Raji, and others. We provide several examples of these applications, one of which gives a new formula for certain families of Hurwitz class numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-18T15:51:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P81",
      "11P99",
      "11E41"
    ],
    "keywords": [
      "core partitions",
      "distinct odd parts",
      "applications",
      "core self-conjugate partitions",
      "hurwitz class numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}