Rational associahedra and noncrossing partitions

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 {\sf rational associahedron}. It is a pure simplicial complex of dimension a-2, and its maximal faces are counted by the {\sf rational Catalan number} \Cat(x)=\Cat(a,b):=\frac{(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 {\sf rational Narayana numbers}) and f-vector (the {\sf rational Kirkman numbers}). We define \Ass(a,b) via {\sf rational Dyck paths}: lattice paths from (0,0) to (b,a) staying above the line y = \frac{a}{b}x. 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.