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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)

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles… (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)

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of N sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles… (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)

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. In this paper we prove… (More)

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It has been cited as a model for traffic flow and protein synthesis. In its most general form, particles may enter and exit at the left with… (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)

A self-avoiding walk (saw) is a path on a lattice that does not pass through the same point twice. Though mathematicians have studied saws for over fifty years, the number of n-step saws is unknown. This paper examines a special case of this problem, finding the number of n-step " upside " saws (ussaws), saws restricted to moving up and sideways. It… (More)