An explicit derivation of the Mobius function for Bruhat order

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.Comment: 9 pages; final versio

Teaching models and local‐area networks

The thesis of this paper is that new advances in both microtechnology and LAN technology can now provide teachers with flexible and exciting instructional tools which allow for a powerful integration of teaching model, curriculum content and technology. The first section describes some of the current applications of school‐based LANs. The second section discusses various teaching models, and describes an in‐depth example of how a teacher may go about providing instruction by combining a LAN and these models. The third section addresses the feasibility of such an instructional approach

Abacus models for parabolic quotients of affine Weyl groups

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 also describe several combinatorial models for these parabolic quotients that generalize classical results in affine type A related to core partitions.Comment: 28 pages, To appear, Journal of Algebra. Version 2: Updated with referee's comment

The enumeration of fully commutative affine permutations

We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending 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. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.Comment: 18 pages; final versio

Leading coefficients of Kazhdan--Lusztig polynomials for Deodhar elements

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. The Deodhar elements have previously been characterized using pattern avoidance by Billey--Warrington (2001) and Billey--Jones (2007). In type $A$, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar's (1990) algorithm, we provide some combinatorial criteria to determine when $\mu(x,w) = 1$ for such permutations $w$.Comment: 28 page