# Combinatorics of the three-parameter PASEP partition function - Mathematics > Combinatorics

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Abstract: We consider a partially asymmetric exclusion process PASEP on a finitenumber of sites with open and directed boundary conditions. Its partitionfunction was calculated by Blythe, Evans, Colaiori, and Essler. It is known tobe a generating function of permutation tableaux by the combinatorialinterpretation of Corteel and Williams.We prove bijectively two new combinatorial interpretations. The first one isin terms of weighted Motzkin paths called Laguerre histories and is obtained byrefining a bijection of Foata and Zeilberger. Secondly we show that thispartition function is the generating function of permutations with respect toright-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, byrefining a bijection of Francon and Viennot.Then we give a new formula for the partition function which generalizes theone of Blythe and al. It is proved in two combinatorial ways. The first proof isan enumeration of lattice paths which are known to be a solution of the MatrixAnsatz of Derrida and al. The second proof relies on a previous enumeration ofrook placements, which appear in the combinatorial interpretation of a relatednormal ordering problem. We also obtain a closed formula for the moments ofAl-Salam-Chihara polynomials.

Author: ** Matthieu Josuat-Vergès**

Source: https://arxiv.org/