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{\bf Matthieu Josuat-Verg\`es}
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{\bf Combinatorics of the Three-Parameter PASEP Partition Function}
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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 Françon 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.
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