Binary operations on pattern-avoiding cycles
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909603495673856 |
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| author | Archer, Kassie Graves, Christina Laudone, Robert |
| author_facet | Archer, Kassie Graves, Christina Laudone, Robert |
| contents | Suppose $c_n(σ)$ denotes the number of cyclic permutations in $\mathcal{S}_n$ that avoid a pattern $σ$. In this paper, we define partial groupoid structures on cyclic pattern-avoiding permutations that allow us to build larger cyclic pattern-avoiding permutations from smaller ones. We use this structure to find recursive lower bounds on $c_n(σ)$. These bounds imply that $c_n(σ)$ has a growth rate of at least 3 for $σ\in\{231,312,321\}$ and a growth rate of at least 2.6 for $σ\in\{123,132,213\}$. In the process, we prove (and sometimes improve) a conjecture of Bóna and Cory that $c_n(σ)\geq 2 c_{n-1}(σ)$ for all $σ\in\mathcal{S}_3\setminus\{123\}$ and $n\geq 2.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Binary operations on pattern-avoiding cycles Archer, Kassie Graves, Christina Laudone, Robert Combinatorics Suppose $c_n(σ)$ denotes the number of cyclic permutations in $\mathcal{S}_n$ that avoid a pattern $σ$. In this paper, we define partial groupoid structures on cyclic pattern-avoiding permutations that allow us to build larger cyclic pattern-avoiding permutations from smaller ones. We use this structure to find recursive lower bounds on $c_n(σ)$. These bounds imply that $c_n(σ)$ has a growth rate of at least 3 for $σ\in\{231,312,321\}$ and a growth rate of at least 2.6 for $σ\in\{123,132,213\}$. In the process, we prove (and sometimes improve) a conjecture of Bóna and Cory that $c_n(σ)\geq 2 c_{n-1}(σ)$ for all $σ\in\mathcal{S}_3\setminus\{123\}$ and $n\geq 2.$ |
| title | Binary operations on pattern-avoiding cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.04456 |