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, in the language of two total orders, the probability p₍, that a uniform random 231-avoiding permutation of size n satisfies admits a limit as n is large. Moreover, we establish two further results about the behavior and value of p₍,: (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.