{
  "id": "2103.12875",
  "version": "v1",
  "published": "2021-03-23T22:47:52.000Z",
  "updated": "2021-03-23T22:47:52.000Z",
  "title": "Chain decompositions of q,t-Catalan numbers III: tail extensions and flagpole partitions",
  "authors": [
    "Seongjune Han",
    "Kyungyong Lee",
    "Li Li",
    "Nicholas A. Loehr"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit $k$ into a disjoint union of chains $\\mathcal{C}_{\\mu}$ indexed by partitions of size $k$. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\\textrm{Cat}_n(q,t)=\\textrm{Cat}_n(t,q)$. Previously, we introduced a map NU that builds the tail part of each chain $\\mathcal{C}_{\\mu}$. Our first main contribution here is to extend $\\NU$ and construct larger second-order tails for each chain. Second, we introduce new classes of partitions (flagpole partitions and generalized flagpole partitions) and give a recursive construction of the full chain $\\mathcal{C}_{\\mu}$ for generalized flagpole partitions $\\mu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-23T22:47:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A19",
      "05A17",
      "05E05"
    ],
    "keywords": [
      "t-catalan numbers",
      "chain decompositions",
      "tail extensions",
      "generalized flagpole partitions",
      "construct larger second-order tails"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}