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Matrix Ansatz, lattice paths and rook placements

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 $n^{\mathrm{th}}$ moment of certain $q$-Laguerre polynomials.
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S. Corteel, M. Josuat-Vergès, T. Prellberg, M. Rubey. Matrix Ansatz, lattice paths and rook placements. 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), 2009, Hagenberg, Austria. pp.313-324, ⟨10.46298/dmtcs.2751⟩. ⟨hal-01185444⟩



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