On colored set partitions of type $B_n$

TL;DR

This paper introduces colored set partitions of type $B_n$, deriving formulas for their enumeration, statistical properties, and proving asymptotic normality of the number of non-zero blocks.

Contribution

It generalizes Reiner's type $B_n$ partitions by coloring elements and provides exact and asymptotic formulas for their enumeration and statistical behavior.

Findings

Exact formulas for expectation and variance of non-zero blocks.

Asymptotic expression for total number of colored $B_n$-partitions.

Proof of asymptotic normality of the number of non-zero blocks.

Abstract

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 $O(n^{-1/2}\log^{7/2}{n})$, and prove that the centralized and normalized number of non-zero-blocks is asymptotic normal over colored $B_n$-partitions.