A logical limit law for $231$-avoiding permutations

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.

15 pages; version 3 is the final version, ready for publication in DMTCS

References in corpus (3)

• Logical limit laws for layered permutations and related structures
• Logical limit laws for Mallows random permutations
• First-order convergence for $321$-avoiding permutations

Cited by in corpus (1)

• Random expansions of finite structures with bounded degree