On colored set partitions of type B n

Generalizing Reiner’s notion of set partitions of type B _n , we define colored B _n -partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored B _n -partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored B _n -partition. We find an asymptotic expression of the total number of colored B _n -partitions up to an error of O(n ^?1/2log^7/2 n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored B _n -partitions.