Combinatorics of the three-parameter PASEP partition function

TL;DR

This paper explores the combinatorial structures underlying the PASEP partition function, providing new interpretations, formulas, and connections to permutation statistics, lattice paths, and orthogonal polynomials.

Contribution

It introduces two new combinatorial interpretations of the PASEP partition function and derives a generalized formula with two combinatorial proofs.

Findings

New interpretation via Laguerre histories and Motzkin paths

Partition function as a permutation generating function with specific statistics

A generalized formula for the partition function with combinatorial proofs

Abstract

We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generating function of permutation tableaux by the combinatorial interpretation of Corteel and Williams. We prove bijectively two new combinatorial interpretations. The first one is in terms of weighted Motzkin paths called Laguerre histories and is obtained by refining a bijection of Foata and Zeilberger. Secondly we show that this partition function is the generating function of permutations with respect to right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection of Francon and Viennot. Then we give a new formula for the partition function which generalizes the one of Blythe & al. It is proved in two combinatorial ways. The first proof is an enumeration of lattice paths which are known to be a solution of the Matrix Ansatz of Derrida & al. The second proof relies on a previous enumeration of rook placements, which appear in the combinatorial interpretation of a related normal ordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.