EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model

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Selig, T., Smith, J.P. ORCID: 0000-0002-4209-1604 and Steingrímsson, E., 2018. EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model. Electronic Journal of Combinatorics, 25 (3): P3.14. ISSN 1077-8926

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Official URL: https://doi.org/10.37236/7480

Abstract

A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding uniquely to an acyclic orientation, with a unique sink, of a certain bipartite graph called a Ferrers graph. We give a bijective proof of a result of Ehrenborg and van Willigenburg showing that EW-tableaux of a given shape are equinumerous with permutations with a given set of excedances. This leads to an explicit bijection between EW-tableaux and the much studied Le-tableaux, as well as the tree-like tableaux introduced by Aval, Boussicault and Nadeau.

We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1 correspondence with the minimal recurrent configurations of the Abelian sandpile model on the corresponding Ferrers graph.

Another bijection between EW-tableaux and tree-like tableaux, via spanning trees on the corresponding Ferrers graphs, connects the tree-like tableaux to the minimal recurrent configurations of the Abelian sandpile model on these graphs. We introduce a variation on the EW-tableaux, which we call NEW-tableaux, and present bijections from these to Le-tableaux and tree-like tableaux. We also present results on various properties of and statistics on EW-tableaux and NEW-tableaux, as well as some open problems on these.

Item Type:

Journal article

Publication Title:

Electronic Journal of Combinatorics

Creators:

Selig, T., Smith, J.P. and Steingrímsson, E.

Publisher:

Electronic Journal of Combinatorics

Date:

27 July 2018

Volume:

Number: