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. © 2023, Krasovskii Institute of Mathematics and Mechanics. All rights reserved.