### 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 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 language | English (US) |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 20 |

Issue number | 3 |

State | Published - Sep 26 2013 |

### Fingerprint

### Keywords

- Associahedron
- Catalan number
- Lattice path
- Noncrossing partition

### ASJC Scopus subject areas

- Geometry and Topology
- Theoretical Computer Science
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*20*(3).

**Rational associahedra and noncrossing partitions.** / Armstrong, Drew; Rhoades, Brendon; Williams, Nathan.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 20, no. 3.

}

TY - JOUR

T1 - Rational associahedra and noncrossing partitions

AU - Armstrong, Drew

AU - Rhoades, Brendon

AU - Williams, Nathan

PY - 2013/9/26

Y1 - 2013/9/26

N2 - Each positive rational number x>0 can be written uniquely as x = a / (b - a) for coprime positive integers 00 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.

AB - Each positive rational number x>0 can be written uniquely as x = a / (b - a) for coprime positive integers 00 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.

KW - Associahedron

KW - Catalan number

KW - Lattice path

KW - Noncrossing partition

UR - http://www.scopus.com/inward/record.url?scp=84884685928&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884685928&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84884685928

VL - 20

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

ER -