An algorithm for deciding the finiteness of the number of simple permutations in permutation classes

Frédérique Bassino (1) , Mathilde Bouvel (2, 3) , Adeline Pierrot (4, 5, 6) , Dominique Rossin (7)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 Institut für Mathematik [Zürich]
4 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications
5 Institut für Diskrete Mathematik und Geometrie [Wien]
6 BioInfo - LRI - Bioinformatique (LRI)
7 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

In this article, we describe an algorithm to determine whether a permutation class C given by a finite basis B of excluded patterns contains a finite number of simple permutations. This is a continuation of the work initiated in [Brignall, Ruškuc, Vatter, Simple permutations: decidability and unavoidable substructures, 2008], and shares several aspects with it. Like in this article, the main difficulty is to decide whether C contains a finite number of proper pin-permutations, and this decision problem is solved using automata theory. Moreover, we use an encoding of proper pin-permutations by words over a finite alphabet, introduced by Brignall et al. However, unlike in their article, our construction of automata is fully algorithmic and efficient. It is based on the study of pin-permutations in [6]. The complexity of the overall algorithm is O(n log n+s2k) where n denotes the sum of the sizes of permutations in the basis B, s is the maximal size of a pin-permutation in B and k is the number of pin-permutations in B.

Mots clés: Permutation class; finite basis; simple permutation; algorithm; automaton; pin-permutation

Domaines: Combinatoire [math.CO]

Frédérique Bassino, Mathilde Bouvel, Adeline Pierrot, Dominique Rossin. An algorithm for deciding the finiteness of the number of simple permutations in permutation classes. Advances in Applied Mathematics, 2015, 64, pp.124 - 200. ⟨10.1016/j.aam.2014.12.001⟩. ⟨hal-01818308⟩