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.1. The top row of T has a 1 in every cell. 2. Every other row has at least one cell containing a 0. 3. No four cells of T that form the corners of a rectangle have 0s in two diagonally opposite corners and 1s in the other two.The size of a EW-tableau is one less than the sum of its number of rows and number of columns.An example of a EW-tableau is shown in Figure 2.1.Remark 1.2. Ehrenborg and van Willigenburg considered colorings corresponding to tableaux that in our definition would have a unique fixed, but arbitrary, row of all 1s, and no column with all 0s. As all our results naturally restrict to tableaux of a fixed shape, it is irrelevant where this fixed all-1s row is chosen, and as it facilitates the presentation of our results, we choose to always make this the top row. That implies, of course, that no column has all 0s. The choice of this row being arbitrary is directly linked to the fact, proved by Greene and Zaslavsky [16, Thm. 7.3], that the number of acyclic orientations, with a unique sink, of a graph is independent of which vertex is chosen as the unique sink. In fact, it is easy to show that given an acyclic orientation with a unique

We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of which corresponds to a canonical toppling of the corresponding configurations.

A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs. Date

We introduce a new model of a stochastic sandpile on a graph G containing a sink. When unstable, a site sends one grain to each of its neighbours independently with probability p ∈ (0, 1]. For p = 1, this coincides with the standard Abelian sandpile model. In general, for p ∈ (0, 1), the set of recurrent configurations of this sandpile model is different from that of the Abelian sandpile model. We give a characterisation of this set in terms of orientations of the graph G. We also define the lacking polynomial L G as the generating function counting this set according to the number of grains, and show that this polynomial satisfies a recurrence which resembles that of the Tutte polynomial.