On Singletons and Adjacencies of Set Partitions of Type B

TL;DR

This paper demonstrates that in type B set partitions without zero-block, the joint distribution of singleton pairs and adjacency pairs is symmetric, extending known results from ordinary partitions to type B.

Contribution

It establishes a symmetry property for the joint distribution of singleton and adjacency pairs in type B partitions, generalizing previous results.

Findings

Joint distribution symmetry proven for type B partitions without zero-block

Extends Callan's symmetry result from ordinary to type B partitions

Provides combinatorial insight into structure of type B partitions

Abstract

We show that the joint distribution of the number of singleton pairs and the number of adjacency pairs is symmetric over the set partitions of type $B_n$ without zero-block, in analogy with the result of Callan for ordinary partitions.