### 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) |
---|---|

Pages (from-to) | 933-944 |

Number of pages | 12 |

Journal | Discrete Mathematics and Theoretical Computer Science |

State | Published - 2013 |

### Fingerprint

### 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)

### Cite this

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

**Rational catalan combinatorics : The associahedron.** / Armstrong, Drew; Rhoades, Brendon; Williams, Nathan.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Rational catalan combinatorics

T2 - The associahedron

AU - Armstrong, Drew

AU - Rhoades, Brendon

AU - Williams, Nathan

PY - 2013

Y1 - 2013

N2 - 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) .

AB - 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) .

KW - Associahedron

KW - Dyck path

KW - F-vector

KW - H-vector

KW - Noncrossing partition

KW - Shelling

UR - http://www.scopus.com/inward/record.url?scp=84887480267&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887480267&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84887480267

SP - 933

EP - 944

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

ER -