A simple bijection between permutation tableaux and permutations

We present a simple a bijection between permutations of $\{1,..., n\}$ with $k$ descents and permutation tableaux of length $n$ with $k$ columns

Overpartitions, lattice paths and Rogers-Ramanujan identities

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by multiplicity conditions on the parts. This leads to many new partition and overpartition identities, and provides a unification of a number of well-known identities of the Rogers-Ramanujan type. Among these are Gordon's generalization of the Rogers-Ramanujan identities, Andrews' generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's theorems for overpartitions.

Tableaux combinatorics for the asymmetric exclusion process

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 may enter from the left with probability $\alpha$ and exit from the right with probability $\beta$. In this paper we prove a close connection between the PASEP and the combinatorics of permutation tableaux. (These tableaux come indirectly from the totally nonnegative part of the Grassmannian, via work of Postnikov, and were studied in a paper of Steingrimsson and the second author.) Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration $\tau$ is essentially the generating function for permutation tableaux of shape $\lambda(\tau)$ enumerated according to three statistics. The proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on the {\it matrix ansatz} for the PASEP model. As an application, we prove some monotonicity results for the PASEP. We also derive some enumerative consequences for permutations enumerated according to various statistics such as weak excedence set, descent set, crossings, and occurences of generalized patterns.Comment: Clarified exposition, more general result, new author (SC), 19 pages, 6 figure

An iterative-bijective approach to generalizations of Schur's theorem

We start with a bijective proof of Schur's theorem due to Alladi and Gordon and describe how a particular iteration of it leads to some very general theorems on colored partitions. These theorems imply a number of important results, including Schur's theorem, Bressoud's generalization of a theorem of G\"ollnitz, two of Andrews' generalizations of Schur's theorem, and the Andrews-Olsson identities.Comment: 16 page

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Introduced in the late 1960's, the asymmetric exclusion process (ASEP) 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 with open boundaries. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work for the ASEP with parameters gamma=delta=0. Using our first result and also results of Uchiyama-Sasamoto-Wadati, we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.Comment: An announcement of these results appeared here: http://www.pnas.org/content/early/2010/03/25/0909915107.abstract This version of the paper has updated references and corrects a gap in the proof of Proposition 6.11 which was in the published versio