Authors: Mireille Bousquet-Mélou ¹; Anders Claesson ²; Mark Dukes ^3,4; Sergey Kitaev ⁵

• 1 Laboratoire Bordelais de Recherche en Informatique
• 2 The Mathematics Institute, Reyjavik University
• 3 Science Institute, University of Iceland
• 4 University of Iceland [Reykjavik]
• 5 Institute of Mathematics

We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-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 $\textit{ascent sequences}$. 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.