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

Abstract:
We derive the continued fraction form of the generating function of some new $q$-analogs of the Eulerian numbers $\hat{E}_{k,n}(q)$ introduced by Lauren Williams building on work of Alexander Postnikov. They are related to the number of alignments and weak exceedances of permutations. We show how these numbers are related to crossing and generalized patterns of permutations We generalize to the case of decorated permutations. Finally we show how these numbers appear naturally in the stationary distribution of the ASEP model.

Abstract:
We introduce the notion of crossings and nestings of a permutation. We compute the generating function of permutations with a fixed number of weak exceedances, crossings and nestings. We link alignments and permutation patterns to these statistics. We generalize to the case of decorated permutations. Finally we show how this is related to the stationary distribution of the Partially ASymmetric Exclusion Process (PASEP) model.

Abstract:
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 with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey-Wilson polynomials, a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey-Wilson polynomials can be viewed as a specialization of the multivariate Macdonald-Koornwinder polynomials (also known as Koornwinder polynomials), which in turn give rise to the Macdonald polynomials associated to any classical root system via a limit or specialization. In light of the fact that Koornwinder polynomials generalize the Askey-Wilson polynomials, it is natural to ask whether one can find a particle model whose partition function is related to Koornwinder polynomials. In this article we answer this question affirmatively, by showing that the "homogeneous" Koornwinder moments at q=t recover the partition function for the two-species exclusion process. We also provide a "hook length" formula for Koornwinder moments when q=t=1.

Abstract:
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."

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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. It has been observed that the (unique) stationary distribution of the PASEP has remarkable connections to combinatorics -- see for example the papers of Derrida, Duchi and Schaeffer, and Corteel. Most recently we proved that in fact the (normalized) probability of being in a particular state of the PASEP can be viewed as a certain weight generating function for permutation tableaux of a fixed shape. (This result implies the previous combinatorial results.) However, our proof relied on the matrix ansatz of Derrida et al, and hence did not give an intuitive explanation of why one should expect the steady state distribution of the PASEP to involve such nice combinatorics. In this paper we define a Markov chain -- which we call the PT chain -- on the set of permutation tableaux which projects to the PASEP in a very strong sense. This gives a new proof of our previous result which bypasses the matrix ansatz altogether. Furthermore, via the bijection from permutation tableaux to permutations, the PT chain can also be viewed as a Markov chain on the symmetric group. Another nice feature of the PT chain is that it possesses a certain symmetry which extends the "particle-hole symmetry" of the PASEP. More specifically, this is a graph-automorphism on the state diagram of the PT chain which is an involution; this has a simple description in terms of permutations.