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Convergence law for 231-avoiding permutations


We prove that the class of 231-avoiding permutations satisfies a convergence 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 behaviour 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.

Dates and versions

hal-03908625 , version 1 (20-12-2022)



Michael Albert, Mathilde Bouvel, Valentin Féray, Marc Noy. Convergence law for 231-avoiding permutations. 2022. ⟨hal-03908625⟩
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