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Abstract
A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete nonambiguous binary trees introduced by Aval et al. [2], and introduce a multirooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs.
Original language  English 

Article number  P3.29 
Number of pages  25 
Journal  The Electronic Journal of Combinatorics 
Volume  26 
Issue number  3 
Publication status  Published  16 Aug 2019 
Keywords
 Abelian sandpile model
 permutation graphs
 bijection
 combinatorics
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 2 Finished

The Möbius function of the poset of permutations
EPSRC (Engineering and Physical Sciences Research Council)
29/09/15 → 28/09/18
Project: Research

New combinatorial perspectives on the abelian sandpile model
Steingrimsson, E. & Dukes, M.
EPSRC (Engineering and Physical Sciences Research Council)
1/07/15 → 31/08/18
Project: Research