Séminaire Lotharingien de Combinatoire, 78B.61 (2017), 12 pp.
Enumerative Properties of Grid Associahedra
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.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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