We explain how to enumerate the set of affine permutations that avoid a given permutation pattern by counting points in linear polyhedra. As a result, the counting sequences (with respect to the Coxeter length statistic) always satisfy a linear constant-coefficient recurrence. We also investigate when the counting sequence is periodic.

An n-core partition is an integer partition whose diagram contains no hook lengths equal to n. We prove that the self-conjugate (2n, 2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C_n, generalizing a result of Fishel--Vazirani for type A. We also introduce a major statistic on simultaneous (n, n+1)-core partitions and on self-conjugate simultaneous (2n, 2n+1)-core partitions that yield q-analogues of the type A and type C Coxeter-Catalan numbers.

Bousquet-Melou & Eriksson's lecture hall theorem generalizes Euler's celebrated distinct-odd partition theorem. We present an elementary and short proof of the lecture hall theorem using a simple bijection involving abacus diagrams.

We study the solitaire mancala game Tchoukaillon, including a new description of all winning Tchoukaillon boards with a given length, and an algorithm to reconstruct a complete winning Tchoukaillon board from partial information. We also propose a graph-theoretic generalization of Tchoukaillon for further study.

We study a parameterized addition operation on n-bit binary numbers that interpolates between ordinary addition (mod 2^n) and bitwise addition (in Z_2^n). We then use this truncated operation in place of ordinary addition in several add-rotate-xor hash algorithms that have appeared as finalists in the NIST competition. We describe a theoretical attack and a new robustness metric for this class of algorithms.

We give two constructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky.

We introduce abacus diagrams that describe minimal length coset representatives in affine Weyl groups of types B, C, and D. These abacus diagrams use a realization of the affine Weyl group of type C due to Eriksson to generalize a construction of James for the symmetric group.

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.

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.

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.

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.

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.

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.

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.

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.