A method for the enumeration of various classes of column-convex polygons
Discrete Mathematics
On the sandpile group of dual graphs
European Journal of Combinatorics
Enumeration of (p, q)-parking functions
Discrete Mathematics
A proof of the q, t-Catalan positivity conjecture
Discrete Mathematics
The sand-pile model and Tutte polynomials
Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
A family of bijections between G-parking functions and spanning trees
Journal of Combinatorial Theory Series A
(2+2)-free posets, ascent sequences and pattern avoiding permutations
Journal of Combinatorial Theory Series A
Partition and composition matrices
Journal of Combinatorial Theory Series A
The Art of Computer Programming: Combinatorial Algorithms, Part 1
The Art of Computer Programming: Combinatorial Algorithms, Part 1
Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers
Journal of Combinatorial Theory Series A
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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 an mxn 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 et al. (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)xn rectangle.