### 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 language | English (US) |
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Pages (from-to) | 933-944 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - Nov 18 2013 |

Event | 25th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2013 - Paris, France Duration: Jun 24 2013 → Jun 28 2013 |

### Keywords

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

### ASJC Scopus subject areas

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

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## Cite this

*Discrete Mathematics and Theoretical Computer Science*, 933-944.