Sandpiles and order structure of integer partitions

In this paper, we study the orders obtained by the generalized dynamics of the sand piles model (SPM). We show that these orders are suborders of L _B, lattice of integer partitions introduced in Brylawski (Discrete Math. 6 (1973) 201), and we deduce from that a characterization of their fixed point. We prove that these orders form an increasing sequence of lattices from SPM to L _B. We then characterize longest paths in these lattices and give a formula describing their length.