In this paper, we give the unique affine crystal structure for the type E_6^{(1)} Kirillov-Reshetikhin crystals that correspond to the leaf nodes of the finite E_6 Dynkin diagram. We also introduce a tableau model for classical highest weight crystals of type E that generalizes a construction of Kashiwara--Nakashima.

In this paper, we give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length. This extends formulas due to Stembridge and Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. Chris has some code available on his website.

In this updated paper, we use heaps to show that the generating functions for certain fully-commutative pattern classes can be rationally transformed to give generating functions for the corresponding freely-braided and maximally clustered pattern classes. The maximally clustered permutations are characterized by avoiding the classical permutation patterns 3421, 4312, and 4321.

In this paper, we give an explicit nonrecursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Mobius function, recovering a classical result due to Verma.

In this paper, we describe a bijection between $\ell$-core partitions with first part $k$ and $(\ell-1)$-core partitions with first part less than or equal to $k$, for fixed positive integers $\ell$ and $k$. This bijection has a geometric interpretation in terms of the root lattice of type $A_{\ell-1}$ as well as a natural description in terms of another correspondence due to Lapointe--Morse.

In this paper, we show that the leading coefficient of the Kazhdan--Lusztig polynomial $P_{x,w}(q)$ known as $\mu(x,w)$ is always either 0 or 1 when w is a Deodhar element of a finite Weyl group.

In this paper, 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. We also briefly discuss the application of heaps to permutation pattern characterization.

In this paper, we define embedded factor pattern-avoidance for general Coxeter groups, and use it to characterize when Deodhar's (1990) algorithm yields a simple combinatorial formula for the Kazhdan-Lusztig polynomials of finite Weyl groups.