Some Families of Type $B$ Set Partitions Counted by the Dowling Numbers

TL;DR

This paper explores specific classes of type B set partitions, their enumeration via Dowling numbers, and their combinatorial properties, including real-rootedness and descent statistics, with implications for Gamma-positivity.

Contribution

It establishes bijections between classes of type B partitions, proves real-rootedness of their generating polynomials, and analyzes descent statistics with new combinatorial insights.

Findings

Classes are in bijection with type B partitions.

Block-generating polynomials are real-rooted.

Descent distribution is Gamma-positive and homomesic.

Abstract

In this paper, we study type $B$ set partitions without zero block. Certain classes of these partitions, such as merging-free and separated partitions (enumerated by the Dowling numbers), are investigated. We show that these classes are in bijection with type $B$ set partitions. The intersection of these two classes is also studied, and we prove that their block-generating polynomials are real-rooted. Finally, we study the descent statistics on the class of permutations obtained by flattening type $B$ merging-free partitions. Using the valley-hopping action, we prove the Gamma-positivity of the descent distribution and provide a combinatorial interpretation of the Gamma-coefficients. We also show that the descent statistic is homomesic under valley-hopping.