Enumerative properties of Grid-Associahedra

Abstract

We continue the study of the nonkissing complex that was introduced by Petersen, Pylyavskyy, and Speyer and was studied lattice-theoretically by the second author. We introduce a theory of Grid-Catalan combinatorics, given the initial data of a nonkissing complex, and show how this theory parallels the well-known Coxeter-Catalan combinatorics. In particular, we present analogues of Chapoton's F-triangle, H-triangle, and M-triangle and give combinatorial, lattice-theoretic, and geometric interpretations of the objects defining these. In our Grid-Catalan setting, we prove that Chapoton's F-triangle and H-triangle identity holds, and we conjecture that Chapoton's F-triangle and M-triangle identity also holds. As an application, we obtain a bijection between the facets of the nonkissing complex and of the noncrossing complex, which provides a partial solution to an open problem of Santos, Stump, and Welker.