Sand Piles Models of Signed Partitions with d Piles

Let r,d ≤ n be nonnegative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partition that we denote by S(n,d,r). A generic element of this model is a signed integer partition with exactly d all distinct nonzero parts, whose maximum positive summand is not exceeding r and whose minimum negative summand is not less than -(n-r). In particular, we determine the covering relations, the rank function, and the parallel convergence time from the bottom to the top of S(n,d,r) by using an abstract Sand Piles Model with three evolution rules. The lattice S(n,d,r) was introduced by the first two authors in order to study some combinatorial extremal sum problems.