Minimally Intersecting Set Partitions of Type B

TL;DR

This paper extends the concept of minimally intersecting set partitions to type B, providing formulas for counting such partitions and their variants, and relating these to Dowling numbers and Dobinski's formula.

Contribution

It introduces formulas for counting minimally intersecting type B partitions and their zero-block variants, advancing combinatorial enumeration in this area.

Findings

Derived formulas for minimally intersecting r-tuples of B_n-partitions

Established formulas for partitions without zero-block

Connected results to Dowling numbers and Dobinski's formula

Abstract

Motivated by Pittel's study of minimally intersecting set partitions, we investigate minimally intersecting set partitions of type B. We find a formula for the number of minimally intersecting r-tuples of $B_n$-partitions, as well as a formula for the number of minimally intersecting r-tuples of $B_n$-partitions without zero-block. As a consequence, it follows the formula of Benoumhani for the Dowling number in analogy to Dobinski's formula.