Sand Piles Models of Signed Partitions with Piles

Let be nonnegative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partitions that we denote by . A generic element of this model is a signed integer partition with exactly all distinct nonzero parts, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than . In particular, we determine the covering relations, the rank function, and the parallel convergence time from the bottom to the top of by using an abstract Sand Piles Model with three evolution rules. The lattice was introduced by the first two authors in order to study some combinatorial extremal sum problems. 1. Introduction Discrete dynamical models whose configurations are integer partitions are also called Sand Piles Models and they have been deeply investigated. In these models an integer partition is treated as a sequence of piles of grains of sand and each singular grain as a single integer unit. An evolution rule in these models is a rule which describes how to move some particular grains of a configuration in order to obtain another configuration. The famous Brylawski paper [1] can be considered the first implicit study of an integer partitions lattice by means of two evolution dynamical rules which determine the covering relations of this lattice. In [1] Brylawski proposed a dynamical approach to study the lattice of all the partitions of a fixed positive integer with the dominance order. However, the explicit identification of a specific set of integer partitions with a Sand Piles Model begins in [2, 3]. In the Sand Piles Model introduced by Goles and Kiwi in [3], denoted by , a sand pile is represented by an ordered partition of an integer , that is, a decreasing sequence having sum , and the movement of a sand grain respects the following rule. Rule (vertical rule). One grain can move from a column to the next one if the difference of height of these two columns is greater than or equal to 2. In the model (introduced by Brylawski, 1973 [1]), the movement of a sand grain respects Rule 1 and Rule 2, which is described as follows. Rule (horizontal rule). If a column containing grains, is followed by a sequence of columns containing grains and then one column containing grains, one grain of the first column can slip to the last one. The Sand Piles Model is a special case of the more general Chip Firing Game ( ), which was introduced by Spencer in [4] to study some “balancing game”. There are a lot of specializations and extensions of this model which have been introduced and studied under