# 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 $\mathsf{Ass}(x)=\mathsf{Ass}(a,b)$ 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* $$\mathsf{Cat}(x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)!}{a!\,b!}.$$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 $\mathsf{Ass}(a,b)$ is shellable and give nice product formulas for its $h$-vector (the *rational Narayana numbers*) and $f$-vector (the *rational Kirkman numbers*). We define $\mathsf{Ass}(a,b)$ via *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 $m+1$.