Symmetric Distribution of Crossings and Nestings in Permutations of Type B

TL;DR

This paper extends known symmetric distribution results of crossings and nestings from type A permutations to type B, providing new combinatorial insights into their structure.

Contribution

It proves a type B analogue of the symmetric distribution of crossings and nestings, expanding the understanding of permutation patterns in combinatorics.

Findings

Established a Bn-analogue of symmetric crossing and nesting distributions.

Extended previous type A results to type B permutations.

Contributed to the combinatorial theory of permutation patterns.

Abstract

This note contains two results on the distribution of crossing numbers and nesting numbers in permutations of type B. More precisely, we prove a Bn- analogue of the symmetric distribution of crossings and nestings of permutations due to Corteel (Adv. in Appl. Math., 38(2)(2007), 149-163) as well as the sym- metric distribution of k-crossings and k-nestings of permutations due to Burrill et al. (DMTCS proc. AN, (2010), 461-468).