We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area, bounce) and (area, dinv) give rise to the same q, t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q, t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q, t-Narayana polynomials which relates them to the famous diagonal harmonics.

We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular, we study the representability of its dual, providing an extension of the Gale duality to this setting.Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial. * supported by the

Abstract. We introduce two operators on stable configurations of the sandpile model that provide an algorithmic bijection between recurrent and parking configurations. This bijection preserves their equivalence classes with respect to the sandpile group. The study of these operators in the special case of the complete bipartite graph Km,n naturally leads to a generalization of the well known Cyclic Lemma of Dvoretsky and Motzkin, via pairs of periodic bi-infinite paths in the plane having slightly different slopes. We achieve our results by interpreting the action of these operators as an action on a point in the grid Z 2 which is pointed to by one of these pairs of paths. Our Cyclic lemma allows us to enumerate several classes of polyominoes, and therefore builds on the work of Irving and Rattan (2009), Chapman et al. (2009), and Bonin et al. (2003.

We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description parallels the one given by Orlik and Solomon for arrangements of hyperplanes, and builds on De Concini and Procesi's work on the rational cohomology of unimodular toric arrangements. As a byproduct we extend Dupont's rational formality result to formality over Z.The data needed in order to state the presentation is fully encoded in the poset of connected components of intersections of the arrangement.

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We conjecture a formula for the symmetric function [n−k]t [n]t ∆ hm ∆e n−k ω(pn) in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington [20], recently proved by Sergel [25] after the breakthrough of Carlsson and Mellit [4]. Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson [14], answering one of their questions. We support our conjecture by proving the specialization m = q = 0, reducing it to the same case of the Delta conjecture, and the Schröder case, i.e. the case ·, e n−d h d . The latter provides a broad generalization of the q, t-square theorem of Can and Loehr [3]. We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed m and n. Finally, in the appendix, we give a new proof of the Delta conjecture at q = 0.Partially labelled Dyck paths differ from labelled Dyck paths only in that 0 is allowed as a label in the former and not in the latter.Definition 2.2. We define for each D ∈ PLD(m, n) a monomial in the variables x 1 , x 2 , . . . : we setwhere l i (D) is the label of the i-th vertical step of D (the first being at the bottom). Notice that x 0 does not appear, which explains the word partially.

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We prove that the Ehrhart polynomial of a zonotope is a specialization of the multiplicity Tutte polynomial introduced in [15].