We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled 2+2-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called ascent sequences. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern 3{\bar 1}52{\bar 4}, and enumerate those permutations, thus settling a conjecture of Pudwell.