Sand Piles Models of Signed Partitions with $d$ Piles

Let $r , d\le n$ be non-negative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partitions that we denote by $S(n,d,r)$. A generic element of this model is a signed integer partition with exactly $d$ all distinct non-zero 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.