### Abstract

Original language | English |
---|---|

Article number | P3.29 |

Journal | Electronic Journal of Combinatorics |

Volume | 26 |

Issue number | 3 |

Publication status | Published - 16 Aug 2019 |

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### Keywords

- math.CO

### Cite this

*Electronic Journal of Combinatorics*,

*26*(3), [P3.29].

**Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees.** / Dukes, Mark; Selig, Thomas; Smith, Jason P.; Steingrimsson, Einar.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 26, no. 3, P3.29.

}

TY - JOUR

T1 - Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees

AU - Dukes, Mark

AU - Selig, Thomas

AU - Smith, Jason P.

AU - Steingrimsson, Einar

N1 - Dukes, Selig and Steingr´ımsson were supported by grant EP/M015874/1 from The Engineering and Physical Sciences Research Council. Smith was supported by grant EP/M027147/1 from The Engineering and Physical Sciences Research Council.

PY - 2019/8/16

Y1 - 2019/8/16

N2 - 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 non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted 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.

AB - 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 non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted 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.

KW - math.CO

M3 - Article

VL - 26

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

M1 - P3.29

ER -