Rational catalan combinatorics: The associahedron

Drew Armstrong, Brendon Rhoades, Nathan Williams

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Abstract

Each positive rational number x > 0 can be written uniquely as x = a/(b - A) for coprime positive integers 0 < a < b. We will identify x with the pair (a; b). In this extended abstract we use rational Dyck paths to define for each positive rational x > 0 a simplicial complex Ass(x) = Ass(a; b) called the rational associahedron. It is a pure simplicial complex of dimension a - 2, and its maximal faces are counted by the rational Catalan number Cat(x) = Cat(a, b) := (a + b - 1)/ a! b! : The cases (a, b) = (n, n + 1) and (a, b) = (n, kn + 1) recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that Ass(a, b) is shellable and give nice product formulas for its h-vector (the rational Narayana numbers) and f-vector (the rational Kirkman numbers). We define Ass(a, b) .

Original languageEnglish (US)
Pages (from-to)933-944
Number of pages12
JournalDiscrete Mathematics and Theoretical Computer Science
StatePublished - 2013

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Keywords

  • Associahedron
  • Dyck path
  • F-vector
  • H-vector
  • Noncrossing partition
  • Shelling

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science
  • Computer Science(all)

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