Metrics on permutations with the same peak set
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916098118516736 |
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| author | Diaz-Lopez, Alexander Haymaker, Kathryn Keough, Kathryn Park, Jeongbin White, Edward |
| author_facet | Diaz-Lopez, Alexander Haymaker, Kathryn Keough, Kathryn Park, Jeongbin White, Edward |
| contents | 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 $S_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metrics on permutations with the same peak set Diaz-Lopez, Alexander Haymaker, Kathryn Keough, Kathryn Park, Jeongbin White, Edward Combinatorics 05A05 (Primary) 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 $S_n$. |
| title | Metrics on permutations with the same peak set |
| topic | Combinatorics 05A05 (Primary) |
| url | https://arxiv.org/abs/2401.10719 |