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

TL;DR

This paper proves that the number of block pairs in type B set partitions follows a normal distribution in the limit, extending classical results to a new combinatorial structure using the saddle point method.

Contribution

It establishes the normality of the distribution of block pairs in type B partitions, including those without zero-block, using advanced asymptotic analysis.

Findings

Number of block pairs in type B partitions is normally distributed asymptotically.

Normality also holds for type B partitions without zero-block.

Uses saddle point method for asymptotic analysis.

Abstract

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.