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

Abstract. 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 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 correspon...

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, Sergey Kitaev
• Keywords: Bijections, Certain Integer Sequences, JCT 2010, Permutations