A logical limit law for $231$-avoiding permutations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913294097317888 |
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| author | Albert, Michael Bouvel, Mathilde Féray, Valentin Noy, Marc |
| author_facet | Albert, Michael Bouvel, Mathilde Féray, Valentin Noy, Marc |
| contents | We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $Ψ$, in the language of two total orders, the probability $p_{n,Ψ}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $Ψ$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,Ψ}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_05537 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A logical limit law for $231$-avoiding permutations Albert, Michael Bouvel, Mathilde Féray, Valentin Noy, Marc Combinatorics Probability 05A16, 60C05 (primary), 03C13 (secondary) We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $Ψ$, in the language of two total orders, the probability $p_{n,Ψ}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $Ψ$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,Ψ}$: (i) it is either bounded away from $0$, or decays exponentially fast; (ii) the set of possible limits is dense in $[0,1]$. Our tools come mainly from analytic combinatorics and singularity analysis. |
| title | A logical limit law for $231$-avoiding permutations |
| topic | Combinatorics Probability 05A16, 60C05 (primary), 03C13 (secondary) |
| url | https://arxiv.org/abs/2210.05537 |