Combinatorics of Labelled Parallelogram polyominoes

We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n (the labels) are bijectively attached to the $n$ north steps of the above-bounding path, with the condition that they appear in increasing values along consecutive north steps. We calculate the Frobenius characteristic of the action of the symmetric group S_n on these labels. All these enumeration results are refined to take into account the area of these polyominoes. We make a connection between our enumeration results and the theory of operators for which the intergral Macdonald polynomials are joint eigenfunctions. We also explain how these same polyominoes can be used to explicitly construct a linear basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure

The number of directed k-convex polyominoes

We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed k-convex polyominoes. We show it is a rational function and we study its asymptotic behavior

Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q,t-Narayana polynomial

We classify recurrent configurations of the sandpile model on the complete bipartite graph K_{m,n} in which one designated vertex is a sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m*n rectangle. Several special types of recurrent configurations and their properties via this bijection are examined. For example, recurrent configurations whose sum of heights is minimal are shown to correspond to polyominoes of least area. Two other classes of recurrent configurations are shown to be related to bicomposition matrices, a matrix analogue of set partitions, and (2+2)-free partially ordered sets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. This path bounces off the external edges of the polyomino, and is reminiscent of Haglund's well-known bounce statistic for Dyck paths. We define a collection of polynomials that we call q,t-Narayana polynomials, defined to be the generating function of the bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to the (area,hagbounce) bistatistic defined on Dyck paths in Haglund (2003). In doing so, we have extended a bistatistic of Egge, Haglund, Kremer and Killpatrick (2003) to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and prove this conjecture for numerous special cases. We also show a relationship between Haglund's (area,hagbounce) statistic on Dyck paths, and our bistatistic (area,parabounce) on a sub-collection of those parallelogram polyominoes living in a (n+1)*n rectangle

Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma

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 ${K}_{m,n}$ 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 $\mathbb{Z}^2$ 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).Comment: 28 page

Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers

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 [arXiv:1208.0024]. 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

The new dinv is not so new

In [Duane, Garsia, Zabrocki 2013] the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in [Hicks, Kim 2013] a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in [Haglund, Remmel, Wilson 2018]. Moreover, we give also a non-compositional proof of the "ehh" case of the shuffle conjecture (after [Garsia, Xin, Zabrocki 2014]) by bijectively proving a relation with the two part case of the Delta conjecture