(2+2)-free posets, ascent sequences and pattern avoiding permutations

Mireille Bousquet-Mélou (1) , Anders Claesson (2) , Mark Dukes (3) , Sergey Kitaev (2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 School of Computer Science - The Mathematics Institute, Reyjavik University
3 University of Iceland [Reykjavik]

We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2+2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations defined in terms of a new type of pattern. An attractive property of these patterns is that, like classical patterns, they are closed under the action of D_8, the symmetry group of the square. The fourth class is formed by certain integer sequences, called ascent sequences, which have a simple recursive structure and are shown to encode (2+2)-free posets and permutations. Our bijections preserve numerous statistics. We determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for the class of chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3{\bar 1}52{\bar 4} and use this to enumerate those permutations, thereby settling a conjecture of Pudwell.

Mots clés: pattern-avoiding permutations; 2+2-free posets; bijections; enumeration

Domaines: Combinatoire [math.CO]

Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, Sergey Kitaev. (2+2)-free posets, ascent sequences and pattern avoiding permutations. Journal of Combinatorial Theory, Series A, 2010, 117 (7), pp.884-909. ⟨hal-00353913⟩