Kazhdan--Lusztig polynomials for maximally-clustered hexagon-avoiding permutations

Abstract

We provide a non-recursive description for the bounded admissible sets of masks used by Deodhar's algorithm to calculate the Kazhdan--Lusztig polynomials $P_{x,w}(q)$ of type $A$, in the case when $w$ is hexagon avoiding and maximally clustered. This yields a combinatorial description of the Kazhdan--Lusztig basis elements of the Hecke algebra associated to such permutations $w$. The maximally-clustered hexagon-avoiding elements are characterized by avoiding the seven classical permutation patterns $\{3421,4312,4321,46718235,46781235,56718234,56781234\}$. We also briefly discuss the application of heaps to permutation pattern characterization.

18 pages