Rational associahedra and noncrossing partitions

Drew Armstrong, Brendon Rhoades, Nathan Williams

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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 paper we 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. 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) via rational Dyck paths: lattice paths from (0, 0) to (b, a) staying above the line. We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of [2n]. In the case (a, b)=(n, mn+1), our construction produces the noncrossing partitions of [(m + 1) n] in which each block has size m + 1.

Original languageEnglish (US)
JournalElectronic Journal of Combinatorics
Issue number3
StatePublished - Sep 26 2013


  • Associahedron
  • Catalan number
  • Lattice path
  • Noncrossing partition

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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