Séminaire Lotharingien de Combinatoire, B77h (2018), 48 pp.

Michele D'Adderio and Yvan Le Borgne

The Sandpile Model on Km,n and the Rank of its Configurations

Abstract. We present an algorithm to compute the rank of a configuration of the sandpile model for the complete bipartite graph Km,n of complexity O(m+n). Furthermore, we provide a formula for the generating function of parking sorted configurations on complete bipartite graphs Km,n according to rank, degree, and the sizes m and n. The results in the present paper are similar to those found in a previous paper by Cori and Le Borgne [Electron. J. Combin. 23(1) (2016), Paper 1.31, 47 pp.] for the complete graph Kn+1, and they rely on the analysis of certain operators on the stable sorted configurations of Km,n developed by Aval, D'Adderio, Dukes and Le Borgne in [Adv. Appl. Math. 73 (2016), 59-98].

Received: October 20, 2016. Revised: April 17, 2017. Accepted: January 13, 2017.

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