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

Thomas Selig, Jason P. Smith, Einar Steingrímsson

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

An 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.
LanguageEnglish
Pages1-32
Number of pages32
JournalThe Electronic Journal of Combinatorics
Volume25
Issue number3
Publication statusPublished - 27 Jul 2018

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Sandpile Model
Tableaux
Statistics
Graph in graph theory
Bijection
Diagram
Acyclic Orientation
Configuration
Tableau
Bijective
Open Problems

Keywords

  • EW-tableau
  • Ferrers graph

Cite this

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abstract = "An 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 graphcalled a Ferrers graph. We give a bijective proof of a result of Ehrenborg and vanWilligenburg showing that EW-tableaux of a given shape are equinumerous withpermutations with a given set of excedances. This leads to an explicit bijectionbetween EW-tableaux and the much studied Le-tableaux, as well as the tree-liketableaux introduced by Aval, Boussicault and Nadeau.We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1correspondence 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 spanningtrees 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.",
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EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model. / Selig, Thomas; Smith, Jason P.; Steingrímsson, Einar.

In: The Electronic Journal of Combinatorics, Vol. 25, No. 3, 27.07.2018, p. 1-32.

Research output: Contribution to journalArticle

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