An algorithm computing combinatorial specifications of permutation classes

TL;DR

This paper introduces an automated algorithm to derive combinatorial specifications and generating functions for permutation classes with finite basis and simple permutations, enabling enumeration and random sampling.

Contribution

It provides a fully algorithmic method to compute algebraic systems for permutation classes based on pattern constraints, including both avoidance and containment.

Findings

Generates positive algebraic systems for permutation classes

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 a permutation class C, 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 positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic.