Multiple contractions of permutation arrays

Given a permutation $\sigma$ on $n$ symbols $\{0,1,\dots ,n-1\}$ and an integer $1\le m\le n-1$, the $m$th contraction of $\sigma$ is the permutation ${\sigma }^{{\mathsf{C}\mathsf{T}}^m}$ on $n-m$ symbols obtained by deleting the symbols $n-1,n-2,\dots ,n-m$ from the cycle decomposition of $\sigma$. The Hamming distance $\mathrm{h}\mathrm{d}(\sigma ,\tau )$ between two permutations $\sigma$ and $\tau$ is the number of symbols $x$ such that $\sigma (x)\mathrm{\ne }\tau (x)$. 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 ${\rm hd}(\sigma,\tau)-{\rm hd}(\sigma^{{\sf CT}^m},\tau^{{\sf CT}^m})\leq 2m$.