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

Mark Dukes, Yvan Le Borgne|arXiv (Cornell University)|Jul 31, 2012

Advanced Combinatorial Mathematics17 references7 citations

TL;DR

This paper establishes a bijection between recurrent configurations of the abelian sandpile model on the complete bipartite graph $K_{m,n}$ with a designated sink and decorated parallelogram polyominoes inside an $m \times n$ bounding box. It introduces a $q,t$-Narayana polynomial as the generating function of the bistatistic $(\mathsf{area}, \mathsf{parabounce})$ on these polyominoes, extending Haglund's bounce statistic to a new combinatorial class and proving symmetry in numerous special cases.

ABSTRACT

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.

Motivation & Objective

• To classify recurrent configurations of the abelian sandpile model on the complete bipartite graph $K_{m,n}$ with one designated sink vertex.
• To establish a combinatorial bijection between these recurrent configurations and decorated parallelogram polyominoes bounded by an $m \times n$ rectangle.
• To define and study a new bistatistic $(\mathsf{area}, \mathsf{parabounce})$ on parallelogram polyominoes, analogous to Haglund’s bounce on Dyck paths.
• To introduce and investigate $q,t$-Narayana polynomials as generating functions of this bistatistic, with connections to known combinatorial objects.
• To address a question by Egge et al. (2003) on extending $q,t$-Catalan statistics to new combinatorial families by proposing a natural extension to parallelogram polyominoes.

Proposed method

• Construct a canonical toppling process for recurrent configurations, which induces a path within the associated parallelogram polyomino.
• Define the $\mathsf{parabounce}$ statistic as the number of bounces of this path off the external edges of the polyomino, analogous to Haglund’s bounce on Dyck paths.
• Define the $q,t$-Narayana polynomial $F_{m,n}(q,t)$ as the generating function of the bistatistic $(\mathsf{area}, \mathsf{parabounce})$ over all parallelogram polyominoes in an $m \times n$ bounding box.
• Use the bijection between recurrent configurations and polyominoes to transfer properties of configurations (e.g., minimal sum of heights, almost non-zero configurations) to polyomino structures.
• Prove symmetry of the $q,t$-Narayana polynomials in $q$ and $t$ for numerous special cases, including $F_{m,n}(q,t)$ for small $m,n$.
• Establish a connection between the $\mathsf{parabounce}$ statistic on a subfamily of polyominoes in $(n+1) \times n$ rectangles and Haglund’s $\mathsf{hagbounce}$ statistic on Dyck paths.

Experimental results

Research questions

• RQ1 How can recurrent configurations of the sandpile model on $K_{m,n}$ with a single sink vertex be classified and bijectively related to combinatorial objects?
• RQ2 Can a $q,t$-analogue of the Narayana numbers be constructed via a bistatistic on parallelogram polyominoes, extending the $q,t$-Catalan framework?
• RQ3 Is the $q,t$-Narayana polynomial symmetric in $q$ and $t$, and if so, under what conditions or for which subclasses?
• RQ4 What is the relationship between the $\mathsf{parabounce}$ statistic on parallelogram polyominoes and Haglund’s $\mathsf{hagbounce}$ statistic on Dyck paths?
• RQ5 How do special classes of recurrent configurations—such as minimal or almost non-zero configurations—correspond to known combinatorial structures like bicomposition matrices or (2+2)-free posets?

Key findings

• Recurrent configurations of the sandpile model on $K_{m,n}$ with a sink are in bijection with parallelogram polyominoes in an $m \times n$ bounding box.
• Minimal recurrent configurations (in sum of heights) correspond exactly to ribbon parallelogram polyominoes of minimal area.
• Almost non-zero recurrent configurations on $K_{n,n}$ are in one-to-one correspondence with bicomposition matrices, and upper triangular bicomposition matrices correspond to a special subclass.
• The $q,t$-Narayana polynomial $F_{m,n}(q,t)$ is defined as the generating function of the bistatistic $(\mathsf{area}, \mathsf{parabounce})$ on parallelogram polyominoes, with explicit rational generating functions provided for small $m,n$.
• The polynomial $F_{m,n}(q,t)$ is symmetric in $q$ and $t$ for all tested cases, including $F_{2,2}, F_{3,3}, F_{2,3}, F_{3,4}, F_{4,4}, F_{3,5}, F_{2,6}$, and the authors prove symmetry in numerous special cases.
• A subfamily of parallelogram polyominoes in $(n+1) \times n$ rectangles yields a $q,t$-Narayana polynomial whose $\mathsf{parabounce}$ statistic matches Haglund’s $\mathsf{hagbounce}$ statistic on Dyck paths, establishing a direct link between the two settings.