Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

Abstract

We prove that the noncrossing partition lattices associated with the complex reflection groups $G(d,d,n)$ for $d,n\ge 2$ admit symmetric decompositions into Boolean subposets. As a result, these lattices have the strong Sperner property and their rank-generating polynomials are symmetric, unimodal, and $\gamma$-nonnegative. We use computer computations to complete the proof that every noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus answering affirmatively a question raised by D. Armstrong.

30 pages, 5 figures, 1 table. Final version. The results of the initial version were extended to symmetric Boolean decompositions of noncrossing partition lattices

References in corpus (1)

• Enumerative properties of generalized associahedra

Cited by in corpus (3)

• Gamma-positivity in combinatorics and geometry
• On the Sperner property for the absolute order on complex reflection groups
• A symmetric chain decomposition of $N(m,n)$ of composition