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

TL;DR

This paper establishes bijections between EW-tableaux, Le-tableaux, and tree-like tableaux, linking them to permutations and minimal recurrent configurations of the Abelian sandpile model on Ferrers graphs, revealing deep combinatorial connections.

Contribution

It provides explicit bijections among various tableaux types and connects them to sandpile model configurations, advancing understanding of their combinatorial structures.

Findings

EW-tableaux are equinumerous with permutations with given excedances.

EW-tableaux correspond to minimal recurrent configurations of the Abelian sandpile.

New tableaux variations and open problems are introduced.

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.