Multiple contractions of permutation arrays

TL;DR

This paper investigates how contracting permutations by deleting certain symbols affects their Hamming distance, providing a complete characterization and conditions for the change in distance after contraction.

Contribution

It offers a complete characterization of the impact of a single contraction on the Hamming distance between permutations, a novel insight in permutation analysis.

Findings

Characterization of contraction effects on Hamming distance

Conditions for the change in Hamming distance after contraction

Insights into permutation array contractions

Abstract

Given a permutation $\sigma$ on $n$ symbols $\{0, 1, \ldots, n-1\}$ and an integer $1 \leq m \leq n-1$, the $m$th contraction of $\sigma$ is the permutation $\sigma^{{\sf CT}^m}$ on $n-m$ symbols obtained by deleting the symbols $n-1, n-2, \ldots, n-m$ from the cycle decomposition of $\sigma$. The Hamming distance ${\rm hd}(\sigma,\tau)$ between two permutations $\sigma$ and $\tau$ is the number of symbols $x$ such that $\sigma(x) \neq \tau(x)$. In this paper we give a complete characterization of the effect of a single contraction on the Hamming distance between two permutations. This allows us to obtain sufficient conditions for ${\rm hd}(\sigma,\tau)-{\rm hd}(\sigma^{{\sf CT}^m},\tau^{{\sf CT}^m})\leq 2m$.