An algorithm computing combinatorial specifications of permutation classes

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.