Séminaire Lotharingien de Combinatoire, 86B.8 (2022), 12 pp.
Sergi Elizalde
Walks in Simplices, Cylindric Tableaux, and Asymmetric Exclusion Processes
Abstract.
We establish bijections between three classes of combinatorial objects that have been studied in different contexts: lattice walks in simplicial regions as introduced by Mortimer-Prellberg, standard cylindric tableaux as introduced by Gessel-Krattenthaler and Postnikov, and sequences of states in the totally asymmetric simple exclusion process. This perspective gives new insights into these objects,
providing a vehicle to translate enumerative results from lattice walks to tableaux,
and to interpret symmetries that are natural in one setting (\textit{e.g.}\ conjugation of tableaux) as involutions in another.
Specifically, it allows us to use a cylindric analogue of the Robinson-Schensted correspondence to give an alternative bijective proof of a recent result of Courtiel, Elvey Price and Marcovici relating forward and backward walks in simplices.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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