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arXiv · 2109.13585

Multiple contractions of permutation arrays

Carmen Amarra·Dom Vito A. Briones·Manuel Joseph C. Loquias
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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 ${\rm hd}(\sigma,\tau)-{\rm hd}(\sigma^{{\sf CT}^m},\tau^{{\sf CT}^m})\leq 2m$.

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BibTeXRIS

Carmen Amarra, Dom Vito A. Briones, Manuel Joseph C. Loquias. 2021-09-28. Multiple contractions of permutation arrays. https://doi.org/10.1007/s10801-023-01238-2

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