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.