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

Abstract

We provide a nonrecursive 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.