Rational Associahedra and Noncrossing Partitions

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 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) via rational Dyck paths: lattice paths from (0, 0) to (b, a) staying above the line y = abx. 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.