Matrix Ansatz, lattice paths and rook placements

TL;DR

This paper provides combinatorial interpretations of the Matrix Ansatz for PASEP using lattice paths and rook placements, leading to new enumeration formulas and generating functions for permutations and q-Laguerre polynomials.

Contribution

It introduces two combinatorial interpretations of the Matrix Ansatz, offering new proofs and formulas for the PASEP partition function and related generating functions.

Findings

New enumeration formula for PASEP partition function

Generating functions for permutations by ascents and patterns

Connections to q-Laguerre polynomial moments

Abstract

We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrences of the pattern 13-2, the generating function according to weak exceedances and crossings, and the n-th moment of certain q-Laguerre polynomials.