A logical limit law for $231$-avoiding permutations

We prove that the class of 231-avoiding permutations satisfies a logical limit law, i.e. that for any first-order sentence $\mathrm{\Psi }$, in the language of two total orders, the probability $p_{n,\mathrm{\Psi }}$ that a uniform random 231-avoiding permutation of size $n$ satisfies $\mathrm{\Psi }$ admits a limit as $n$ is large. Moreover, we establish two further results about the behavior and value of $p_{n,\mathrm{\Psi }}$: (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.