  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 $3{\bar 1}52{\bar 4}$ and use this to
enumerate those permutations, thereby settling a conjecture of Pudwell.
