The sandpile model on a bipartite graph, parallelogram polyominoes, and a q,t-Narayana polynomial
| Autor: | Mark Dukes, Le Borgne, Y. |
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| Přispěvatelé: | Department of Computer and Information Sciences [Univ Strathclyde], University of Strathclyde [Glasgow], Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), Le Borgne, Yvan |
| Jazyk: | angličtina |
| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), Jul 2012, Nagoya, Japan. pp.337-348 Scopus-Elsevier |
| Popis: | International audience; We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph Km,n in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m×n rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call q,t-Narayana polynomials, the generating functions of the bistatistic (area, bounceWeight) on the set of parallelogram polyominoes, akin to Haglund's (area, hbounce) bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. 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 discuss the proofs for numerous special cases. We also show a relationship between the q,t-Catalan polynomials and our bistatistic (area, bounceWeight) on a subset of parallelogram polyominoes. |
| Databáze: | OpenAIRE |
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