Lauren K. Williams

Learn More
Lauren Williams (joint work with Einar Steingrímsson) We introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the " Le-diagrams " of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of(More)
Postnikov [7] has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr + k,n , and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in Gr + k,n according(More)
We introduce some combinatorial objects called staircase tableaux, which have cardinality 4(n)n!, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics introduced in the late 1960s, which describes a system of interacting particles hopping left and right on a(More)
In this paper we study the partially ordered set Q J of cells in Rietsch's [20] cell decomposition of the totally nonnegative part of an arbitrary flag variety P J ≥0. Our goal is to understand the geometry of P J ≥0 : Lusztig [13] has proved that this space is contractible, but it is unknown whether the closure of each cell is contractible, and whether P J(More)
We introduce a new family of noncommutative analogs of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux(More)