rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Research on combinatorial statistics
    sub title: crossings and nestings in discrete structures
  name:
    name part: Poznanovikj, Svetlana.
    role:
      role term: creator
  name:
    name part: Yan, Catherine H.
    role:
      role term: thesis advisor.
  type of resource: text
  genre: bibliography
  origin info:
    place:
      place term: txu
    date issued: 2011
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: remote
    extent: 1 online resource.
  abstract: We study the distribution of combinatorial statistics that exhibit a structure of crossings and nesting in various discrete structures, in particular, in set partitions, matchings, and fillings of moon polyominoes with entries 0 and 1. Let pi and y be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integers l, m >̲ 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to pi, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (pi, l) and T (y, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (pi, l) and T (pi, l) for l = 0; 1, then it coincides on T (pi, l, m) and T (y, l, m) for all l, m >̲ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. Moreover, we give a bijection between partially directed paths in the symmetric wedge y = +̲ x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of E. J. Janse van Rensburg, T. Prellberg, and A. Rechnitzer that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = +̲ x with k north steps is equal to the number of matchings on [2n] with k nestings. Furthermore, we propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s, A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classic…
  note: by Svetlana Poznanovikj.
  note: "Major Subject: Mathematics"
  note: Title from author supplied metadata (automated record created 2011-11-01 09:04:20).
  note: Electronic resource.
  note: Doctor of Philosophy Texas A&M University 2010 http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8297
  note: Includes bibliographical references.
  note: Text (Dissertation)
  subject:
    topic: Major mathematics
  subject:
    topic: crossings
  subject:
    topic: set partitions
  subject:
    topic: fillings of moon polyominoes
  subject:
    topic: nestings
  subject:
    topic: major index
  identifier: http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8297
  record info:
    record content source: TXA
    record creation date: 111108
    record change date: 20151202144026.0
    record identifier: in00002724722