A logical limit law for $231$-avoiding permutations

TL;DR

This paper proves that 231-avoiding permutations follow a logical limit law, with probabilities of first-order properties converging and exhibiting specific asymptotic behaviors, using analytic combinatorics techniques.

Contribution

It establishes a logical limit law for 231-avoiding permutations and characterizes the possible limit probabilities, a novel result in combinatorics and logic.

Findings

Probability of properties converges as permutation size grows

Limit probabilities are either bounded away from zero or decay exponentially

Possible limit probabilities are dense in [0,1]

Abstract

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.