Combinatorial of restricted permutations according to the number of crossings

TL;DR

This thesis explores the combinatorial properties of restricted permutations avoiding certain patterns, focusing on the distribution of crossings and establishing bijections that preserve this statistic.

Contribution

It introduces new bijections that relate permutations with pattern restrictions to those with preserved crossing numbers, enriching the combinatorial understanding.

Findings

Identified relationships between crossing distributions in restricted permutations

Constructed bijections preserving the number of crossings

Connected crossings with known combinatorial triangles

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.