2017/11/05 by Thomas Selig, Selig, Thomas, Jason P. Smith +3 · 1 citation
Mathematics · #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1711.01622
A EW-tableau is a certain 0/1-filling of a Ferrers diagram, corresponding\nuniquely to an acyclic orientation, with a unique sink, of a certain bipartite\ngraph called a Ferrers graph. We give a bijective proof of a result of\nEhrenborg and van Willigenburg showing that EW-tableaux of a given shape are\nequinumerous with permutations with a given set of excedances. This leads to an\nexplicit bijection between EW-tableaux and the much studied Le-tableaux, as\nwell as the tree-like tableaux introduced by Aval, Boussicault and Nadeau.\n We show that the set of EW-tableaux on a given Ferrers diagram are in 1-1\ncorrespondence with the minimal recurrent configurations of the Abelian\nsandpile model on the corresponding Ferrers graph.\n Another bijection between EW-tableaux and tree-like tableaux, via spanning\ntrees on the corresponding Ferrers graphs, connects the tree-like tableaux to\nthe minimal recurrent configurations of the Abelian sandpile model on these\ngraphs. We introduce a variation on the EW-tableaux, which we call\nNEW-tableaux, and present bijections from these to Le-tableaux and tree-like\ntableaux. We also present results on various properties of and statistics on\nEW-tableaux and NEW-tableaux, as well as some open problems on these.\n