Enumerative Properties of Grid Associahedra

Thomas McConville

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

Coxeter-Catalan combinatorics places familiar Catalan objects in the context of Coxeter systems. Key examples include triangulations of a polygon, nonnesting partitions, and noncrossing partitions. These objects can be interpreted respectively as clusters of a cluster algebra, antichains in the root poset, and elements of a Coxeter group less than a fixed Coxeter element in the absolute order. In each case, the number of objects in question has a simple formula that depends only on the (finite) Coxeter system from which the objects are defined. A richer enumerative relationship between these objects was conjectured by Chapoton and subsequently proved by several authors. We present a new generalization of these Catalan objects as maximal collections of nonkissing paths in the plane, canonical join representations of elements in the Grid-Tamari order, and the shard intersection order of the Grid-Tamari order. We prove that the nonkissing complex admits a particular fan realization from which one can recover the other structures. We conjecture that this fan is the normal fan of a polytope, called the grid associahedron. Furthermore, we prove that one of the identities among Coxeter-Catalan objects conjectured by Chapoton continues to hold in this setting, and we conjecture that the other identities hold as well.