Combinatorics of the three-parameter PASEP partition function

Combinatorics of the three-parameter PASEPpartition function∗Matthieu Josuat-Verg`esUniversit´e Paris-sud and LRI,91405 Orsay CEDEX, FRANCE.josuat@lri.frMathematics Subject Classifications: 05A15, 05A19, 82B23, 60C05.AbstractWe consider a partially asymmetric exclusion process (PASEP) on a finite num-ber of sites with open and directed boundary conditions. Its partition function wascalculated by Blythe, Evans, Colaiori, and Essler. It is known to be a generatingfunctionofpermutationtableauxbythecombinatorial interpretation ofCorteel andWilliams.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 this partitionfunction is the generating function of permutations with respect to right-to-leftminima, right-to-left maxima, ascents, and 31-2 patterns, by refining a bijection ofFranc¸on and Viennot.Then we give a new formula for the partition function which generalizes theone of Blythe & 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 & al. The second proof relies on a previous enumeration of rookplacements, which appear in the combinatorial interpretation of a related normalordering problem. We also obtain a closed formula for the moments of Al-Salam-Chihara polynomials.1 ...