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

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.