On colored set partitions of type Bn

Abstract

Generalizing Reiner’s notion of set partitions of type Bn, we define colored Bn-partitions by coloring the elements in and not in the zero-block respectively. Considering the generating function of colored Bn-partitions, we get the exact formulas for the expectation and variance of the number of non-zero-blocks in a random colored Bn-partition. We find an asymptotic expression of the total number of colored Bn-partitions up to an error of O(n−1/2log7/2n], and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored Bn-partitions.