  In this thesis, we introduced and carried out a combinatorial study of
permutations that avoid one or two patterns of length 3 according to the
statistic number of crossings. For this purpose, we manipulated a bijection of
Elizalde and Pak and constructed other bijections that preserve the number of
crossings. As results, we found, throughout these bijections, various
relationships on the distributions of the number of crossings on restricted
permutations as well as combinatorial interpretations in terms of the number of
crossings on permutations with forbidden patterns of some well known triangles
in the literature.
