Associahedra via spines

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the classical permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction

Alexander Duality and Rational Associahedra

A recent pair of papers of Armstrong, Loehr, and Warrington and Armstrong, Williams, and the author initiated the systematic study of {\em rational Catalan combinatorics} which is a generalization of Fuss-Catalan combinatorics (which is in turn a generalization of classical Catalan combinatorics). The latter paper gave two possible models for a rational analog of the associahedron which attach simplicial complexes to any pair of coprime positive integers a < b. These complexes coincide up to the Fuss-Catalan level of generality, but in general one may be a strict subcomplex of the other. Verifying a conjecture of Armstrong, Williams, and the author, we prove that these complexes agree up to homotopy and, in fact, that one complex collapses onto the other. This reconciles the two competing models for rational associahedra. As a corollary, we get that the involution (a < b) \longleftrightarrow (b-a < b) on pairs of coprime positive integers manifests itself topologically as Alexander duality of rational associahedra. This collapsing and Alexander duality are new features of rational Catalan combinatorics which are invisible at the Fuss-Catalan level of generality.Comment: 23 page

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 {\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.Comment: 21 pages, 8 figure

Compatibility fans for graphical nested complexes

Graph associahedra are natural generalizations of the classical associahedra. They provide polytopal realizations of the nested complex of a graph $G$, defined as the simplicial complex whose vertices are the tubes (i.e. connected induced subgraphs) of $G$ and whose faces are the tubings (i.e. collections of pairwise nested or non-adjacent tubes) of $G$. The constructions of M. Carr and S. Devadoss, of A. Postnikov, and of A. Zelevinsky for graph associahedra are all based on the nested fan which coarsens the normal fan of the permutahedron. In view of the combinatorial and geometric variety of simplicial fan realizations of the classical associahedra, it is tempting to search for alternative fans realizing graphical nested complexes. Motivated by the analogy between finite type cluster complexes and graphical nested complexes, we transpose in this paper S. Fomin and A. Zelevinsky's construction of compatibility fans from the former to the latter setting. For this, we define a compatibility degree between two tubes of a graph $G$. Our main result asserts that the compatibility vectors of all tubes of $G$ with respect to an arbitrary maximal tubing on $G$ support a complete simplicial fan realizing the nested complex of $G$. In particular, when the graph $G$ is reduced to a path, our compatibility degree lies in $\{-1,0,1\}$ and we recover F. Santos' Catalan many simplicial fan realizations of the associahedron.Comment: 51 pages, 30 figures; Version 3: corrected proof of Theorem 2

Stokes posets and serpent nests

We study two different objects attached to an arbitrary quadrangulation of a regular polygon. The first one is a poset, closely related to the Stokes polytopes introduced by Baryshnikov. The second one is a set of some paths configurations inside the quadrangulation, satisfying some specific constraints. These objects provide a generalisation of the existing combinatorics of cluster algebras and nonnesting partitions of type A.Comment: 24 pages, 12 figure