tailieunhanh - Báo cáo toán học: "Combinatorics of the three-parameter PASEP partition function"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Combinatorics of the three-parameter PASEP partition function. | Combinatorics of the three-parameter PASEP partition function Matthieu Josuat-Verges Universite Paris-sud and LRI 91405 Orsay CEDEX FRANCE. josuat@ Submitted Jan 23 2010 Accepted Jan 10 2011 Published Jan 19 2011 Mathematics Subject Classifications 05A15 05A19 82B23 60C05. 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. 1 Introduction The PASEP partition function The partially asymmetric simple exclusion process also called PASEP is a Markov chain describing the evolution of particles in N sites arranged in a line each site being either Partially supported by the grant ANR08-JCJC-0011. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P22 1 empty or occupied by one particle. Particles may enter the leftmost site at a rate a 0 go out the rightmost site at a rate p 0 hop left

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