Geodesic
A logical limit law for $231$-avoiding permutations
Discrete mathematics & theoretical computer science, Permutation Patterns 2023, Tome 26 (2024) no. 1.

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We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\Psi$, in the language of two total orders, the probability $p_{n,\Psi}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\Psi$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\Psi}$: (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.
DOI : 10.46298/dmtcs.11751
Classification : 05A05 
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Albert, Michael; Bouvel, Mathilde; Féray, Valentin; Noy, Marc. A logical limit law for $231$-avoiding permutations. Discrete mathematics & theoretical computer science, Permutation Patterns 2023, Tome 26 (2024) no. 1. doi : 10.46298/dmtcs.11751. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.11751/

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