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

Thomas Selig (Corresponding Author), Jason P. Smith, Einar Steingrimsson

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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.
Original languageEnglish
Article number#P3.14
JournalElectronic Journal of Combinatorics
Volume25
Issue number3
Publication statusPublished - 27 Jul 2018

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Sandpile Model
Tableaux
Statistics
Bijection
Graph in graph theory
Diagram
Acyclic Orientation
Configuration
Tableau
Bijective
One to one correspondence
Spanning tree
Bipartite Graph

Keywords

  • math.CO
  • 05A19

Cite this

EW-tableaux, Le-tableaux, tree-like tableaux and the Abelian sandpile model. / Selig, Thomas (Corresponding Author); Smith, Jason P.; Steingrimsson, Einar.

In: Electronic Journal of Combinatorics, Vol. 25, No. 3, #P3.14, 27.07.2018.

Research output: Contribution to journalArticle

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AB - 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.

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