Combinatorial specification of permutation classes

TL;DR

This paper introduces an automated method to derive combinatorial specifications for permutation classes based on their basis and simple permutations, enabling algebraic generating functions and uniform sampling.

Contribution

It provides a fully algorithmic approach to obtain combinatorial specifications for permutation classes with finite basis and simple permutations, including generating functions and sampling algorithms.

Findings

Produces positive algebraic systems for generating functions

Enables uniform random sampling of permutations in the class

Automates derivation of combinatorial specifications

Abstract

This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positiveand algebraic. It also yields a uniform random sampler of permutations in C. The method presentedis fully algorithmic.