ON SEQUENCES OF ELEMENTARY TRANSFORMATIONS IN THE INTEGER PARTITIONS LATTICE

Abstract

An integer partition, or simply, a partition is a nonincreasing sequence λ = (λ1,λ2,…) of nonnegative integers that contains only a finite number of nonzero components. The length ℓ(λ) of a partition λ is the number of its nonzero components. For convenience, a partition λ will often be written in the form λ = (λ1,…,λt), where t ≥ ℓ(λ); i.e., we will omit the zeros, starting from some zero component, not forgetting that the sequence is infinite. Let there be natural numbers i,j ∈{1,…,ℓ(λ) + 1} such that (1) λi - 1 ≥ λi+1; (2) λj-1 ≥ λj+1; (3) λi = λj+δ, where δ ≥ 2. We will say that the partition η = (λ1,…,λi - 1,…,λj + 1,…,λn) is obtained from a partition λ = (λ1,…,λi,…,λj,…,λn) by an elementary transformation of the first type. Let λi - 1 ≥ λi+1, where i ≤ ℓ(λ). A transformation that replaces λ by η = (λ1,…,λi-1,λi - 1,λi+1,…) will be called an elementary transformation of the second type. The authors showed earlier that a partition μ dominates a partition λ if and only if λ can be obtained from μ by a finite number (possibly a zero one) of elementary transformations of the pointed types. Let λ and μ be two arbitrary partitions such that μ dominates λ. This work aims to study the shortest sequences of elementary transformations from μ to λ. As a result, we have built an algorithm that finds all the shortest sequences of this type.