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

# Michele D'Adderio and Yvan Le Borgne

# The Sandpile Model on *K*_{m,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 *K*_{m,n} of
complexity *O*(*m*+n). Furthermore, we provide a formula for the generating function of parking sorted configurations on complete bipartite graphs *K*_{m,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
*K*_{n+1}, and they rely on the analysis of certain operators on
the stable sorted configurations of *K*_{m,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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