Sand Piles Models of Signed Partitions with Piles

Abstract

Let r,d ≤ n 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. e lattice was introduced by the �rst two authors in order to study some combinatorial extremal sum problems.

Parallel Rank of Two Sandpile Models of Signed Integer Partitions

“…An analogous statement of the previous theorem was proven in [36] for the sublattice of all the signed integer partitions of ( , , ) having positive and negative parts all distinct between them. Below, we draw the Hasse diagram of the lattice (4, 3, 2) by using the evolution Rules…”

“…For the sake of completeness, we recall that in [36] we have computed the rank of the sublattice ( , ) of all the signed partitions of ( , ) having positive and negative parts all distinct between them.…”

Parallel Dynamical Systems over Graphs and Related Topics: A Survey

“…Besides, in [53], a class of lattices and Boolean functions are employed to study the truth of a mathematical conjecture. Other related interesting works in this research line are [54][55][56][57].…”

Rough Sets for n-Cycles and n-Paths

“…Recently, the impetuous development of computer science has placed new questions about the graph structures. For example, the graphs can be studied in terms of sequential dynamical systems (see [2,3,4,5]), by means of parallel dynamics (see [1]), or also for their analogies with both sequential and parallel dynamics on order structures (see [7,8,9,10,15,17,18,19,20,21]). In this paper we continue a research project started in [22,23], where a simple undirected graph is studied as a particular type of information system.…”