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arXiv · 2401.10719

Metrics on permutations with the same peak set

Alexander Diaz-Lopez·Kathryn Haymaker·Kathryn Keough·Jeongbin Park·Edward White
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Abstract

Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$. Given a permutation $σ=σ_1σ_2 \cdots σ_n \in S_n$, we say it has a peak at index $i$ if $σ_{i-1}<σ_i>σ_{i+1}$. Let $\text{Peak}(σ)$ be the set of all peaks of $σ$ and define $P(S;n)=\{σ\in S_n\, | \,\text{Peak}(σ)=S\}$. In this paper we study the Hamming metric, $\ell_\infty$-metric, and Kendall-Tau metric on the sets $P(S;n)$ for all possible $S$, and determine the minimum and maximum possible values that these metrics can attain in these subsets of $S_n$.

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Alexander Diaz-Lopez, Kathryn Haymaker, Kathryn Keough, Jeongbin Park, Edward White. 2024-01-19. Metrics on permutations with the same peak set. https://arxiv.org/abs/2401.10719

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