Combinatorial of restricted permutations according to the number of crossings

Abstract

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.