  We show that there are $n!$ matchings on $2n$ points without, so called, left
(neighbor) nestings. We also define a set of naturally labeled $(2+2)$-free
posets, and show that there are $n!$ such posets on $n$ elements. Our work was
inspired by Bousquet-M\'elou, Claesson, Dukes and Kitaev [J. Combin. Theory
Ser. A. 117 (2010) 884--909]. They gave bijections between four classes of
combinatorial objects: matchings with no neighbor nestings (due to Stoimenow),
unlabeled $(2+2)$-free posets, permutations avoiding a specific pattern, and so
called ascent sequences. We believe that certain statistics on our matchings
and posets could generalize the work of Bousquet-M\'elou et al.\ and we make a
conjecture to that effect. We also identify natural subsets of matchings and
posets that are equinumerous to the class of unlabeled $(2+2)$-free posets.
  We give bijections that show the equivalence of (neighbor) restrictions on
nesting arcs with (neighbor) restrictions on crossing arcs. These bijections
are thought to be of independent interest. One of the bijections maps via
certain upper-triangular integer matrices that have recently been studied by
Dukes and Parviainen [Electron. J. Combin. 17 (2010) \#R53]
