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

Selig, T, Smith, JP ORCID: https://orcid.org/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

Preview

Text 1390794_a2008_Smith.pdf - Post-print Download (440kB) | Preview

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:	25
Number:	3
ISSN:	1077-8926
Identifiers:	Number Type 10.37236/7480 DOI 1390794 Other
Divisions:	Schools > School of Science and Technology
Record created by:	Linda Sullivan
Date Added:	22 Apr 2021 15:14
Last Modified:	14 Jan 2022 12:21
URI:	https://irep.ntu.ac.uk/id/eprint/42748

Actions (login required)

Edit View

Statistics

Views

Downloads