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

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.