# 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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Cited literature [27 references]

https://hal.inria.fr/hal-01185444
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• HAL Id : hal-01185444, version 1

### Citation

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. ⟨hal-01185444⟩

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