I will present some results from a recently completed project that ties together several objects: restricted in a certain way permutations, $(2+2)$-free partially ordered sets, and a certain class of involutions (chord diagrams). Each of these structures can be encoded by a special sequence of numbers, called ascent sequences, thus providing bijections, preserving numerous statistics, between the objects.\\ \noindent In my talk, I will also discuss the generating function for these classes of objects, as well as a restriction on the ascent sequences that allows to settle a conjecture of Pudwell on permutations avoiding $3\bar{1}52\bar{4}$.\\ \noindent This is joint work with Mireille Bousquet-Melou (Bordeaux), Anders Claesson (Reykjavik University) and Mark Dukes (University of Iceland).