Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi></mml:math>-Narayana polynomial

Dukes, Mark; Borgne, Yvan Le

doi:10.1016/j.jcta.2013.01.004

Cited by 22 publications

(45 citation statements)

References 17 publications

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“…

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 [3]. 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.

…”

mentioning

confidence: 60%

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

Aval

D'Adderio²,

Dukes³

et al. 2014

Journal of Combinatorial Theory, Series A

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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.

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“…

…”

mentioning

confidence: 60%

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

Aval

D'Adderio²,

Dukes³

et al. 2014

Journal of Combinatorial Theory, Series A

Self Cite

View full text Add to dashboard Cite

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.

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“…These graphs are a class of bipartite graphs which include the complete bipartite graphs (corresponding to rectangular Ferrers diagrams). As such, this work can be seen as complementing and extending that of [1,2,11,16].…”

Section: Definitions and Backgroundmentioning

confidence: 90%

“…In this section we recall the notion of canonical topplings for Ferrers graphs, which was introduced in the case of complete bipartite graphs by Dukes and Le Borgne [11]. This notion helps us give explicit descriptions of the inverse of φ T C (in Section 3.3) and of the bijection between EW-tableaux and permutations introduced in [16] (recalled also at the end of this section).…”

Section: Canonical Topplingmentioning

confidence: 99%

The Abelian sandpile model on Ferrers graphs — A classification of recurrent configurations

Dukes

Selig

Smith

et al. 2019

European Journal of Combinatorics

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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.

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“…The paper [10] showed how configurations of the sandpile model on the complete bipartite graph K m,n that are both stable and sorted may be viewed as collections of cells in the plane. By sorted we mean that configuration heights are weakly increasing in each part with respect to vertex indices.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 5 we bring together the results of Sections 2, 3 and 4 while also building on the construction given in [10]. We will represent sandpile configurations on K m,n as bi-infinite pairs of paths in the plane.…”

Section: Introductionmentioning

confidence: 99%

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

Aval

D'Adderio²,

Dukes³

et al. 2016

Advances in Applied Mathematics

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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.

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

Cited by 22 publications

References 17 publications

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

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

The Abelian sandpile model on Ferrers graphs — A classification of recurrent configurations

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

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