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A Determinantal Formula for Catalan Tableaux and TASEP Probabilities
du groupe Combinatoire Énumérative et Algébrique

Date 2014-11-07  10:45-11:45
TitreA Determinantal Formula for Catalan Tableaux and TASEP Probabilities 
RésuméWe present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with alpha's and beta's that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux with a given number of rows, which corresponds to the steady state probability that in the TASEP on a lattice with n sites, precisely k of the sites are occupied by particles. This formula is an alpha and beta generalization of the Narayana numbers. We conclude with some extensions to a two-species PASEP, in which there are particles of type 1 and 2 hopping right with rate 1 and hopping left with rate q, with only type 2 particles being able to enter and exit. We describe certain Catalan-like tableaux for the q=0 case, and two-species alternative tableaux for the q=1 case. 
OrateurOlya Mandelshtam 
UrlDepartment of Mathematics University of California, Berkeley, USA 

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