Symmetric Distribution of Crossings and Nestings in Permutations of Type $B$

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 $B_n$-analogue of the symmetric distribution of crossings and nestings of permutations due to Corteel [Adv. Appl. Math. 38(2)(2007), 149–163] as well as the symmetric distribution of $k$-crossings and $k$-nestings of permutations due to Burrill et al. [DMTCS proc. AN (2010), 593–600].