Seminar: Families of Type B Set Partitions Counted by the Dowling Numbers
Speaker: Fufa Beyene, Kotebe University of Education, Ethiopia
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---which are 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. We present a closed formula for the number of non-crossing merging-free partitions. Furthermore, we show that while type A non-crossing separated partitions are counted by the Motzkin numbers, type B non-crossing separated partitions are counted by the sequence enumerating directed animals. 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 γ-positivity of the descent distribution and provide a combinatorial interpretation of the γ-coefficients.
Keywords: Partitions, Dowling numbers, bijections, Motzkin numbers