论文信息 - Chip-Firing on the Complete Split Graph: Motzkin Words and Tiered Parking Functions

Chip-Firing on the Complete Split Graph: Motzkin Words and Tiered Parking Functions

We highlight some results from studying chip-firing on the the complete split graph [5]. In this work it is shown that recurrent states can be characterised in terms of Motzkin words and can also be characterised in terms of a new type of parking function that we call a tiered parking function. These new parking functions arise by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them.

Mark Dukes

[1] Robert Cori,et al. On the Sandpile Group of Dual Graphs , 2000, Eur. J. Comb..

[2] D. Dhar. Theoretical studies of self-organized criticality , 2006 .

[3] Mark Dukes. The sandpile model on the complete split graph, Motzkin words, and tiered parking functions , 2021, J. Comb. Theory, Ser. A.

[4] Jean-Christophe Aval,et al. Two operators on sandpile configurations, the sandpile model on the complete bipartite graph, and a Cyclic Lemma , 2013, Adv. Appl. Math..

[5] Robert Cori,et al. Enumeration of (p, q)-parking functions , 2002, Discret. Math..

[6] Jason P. Smith,et al. EW-Tableaux, Le-Tableaux, Tree-Like Tableaux and the Abelian Sandpile Model , 2018, Electron. J. Comb..

[7] Jason P. Smith,et al. Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees , 2018, Electron. J. Comb..

[8] Alexander Postnikov,et al. Trees, parking functions, syzygies, and deformations of monomial ideals , 2003 .

[9] Jason P. Smith,et al. The Abelian sandpile model on Ferrers graphs - A classification of recurrent configurations , 2019, Eur. J. Comb..

[10] Mark Dukes,et al. Parallelogram polyominoes, the sandpile model on a complete bipartite graph, and a q, t-Narayana polynomial , 2013, J. Comb. Theory, Ser. A.