The Limiting Distribution of the Number of Block Pairs in Type B Set Partitions

It is a classical result of Harper that the limiting distribution of the number of blocks in partitions of the set $\{1, 2,..., n\}$ is normal. In this paper, using the saddle point method we prove the normality of the limiting distribution of the number of block pairs in set partitions of type $B_n$. Moreover, we obtain that the limiting distribution of the number of block pairs in $B_n$-partitions without zero-block is also normal.