(2+2)-free Posets, Ascent Sequences and Pattern Avoiding Permutations

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 ¯ 152 ¯ 4 and use this to enumerate those permutations , thereby settling a conjecture of Pudwell.