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.

💡 Research Summary

The paper investigates type B set partitions of the signed set ⟨n⟩={0,±1,…,±n} under the restriction that the zero block either does not appear or contains only the element 0. After recalling the classical Dowling numbers Dₙ, which count all type B partitions, the authors focus on the subset Π₀ⁿ of partitions without a zero block and denote its cardinality by wₙ. They show that wₙ is the binomial transform of Dₙ, i.e. wₙ=∑{j=0}ⁿ(−1)^{n−j}C(n,j)D_j, and also obtain the closed form wₙ=∑{k=1}ⁿ2^{n−k}S(n,k), linking wₙ directly to ordinary Stirling numbers of the second kind. A recursion w_{n,k}=2k·w_{n−1,k}+w_{n−1,k−1} is proved, and the associated generating polynomials T_n(x)=∑{k≥0}w{n,k}x^k are shown to be real‑rooted.

Two natural statistics on Π₀ⁿ are introduced: the number of merging blocks (mb) and the number of successions (suc). A block is merging if its minimal element exceeds the maximal element of the preceding block; a succession occurs when consecutive integers (up to sign) lie in the same block. Partitions with mb=0 are called merging‑free, while those with suc=0 are called separated. The authors define a “swap” operation and two maps µ_a and ρ_a that respectively turn a merging block into a succession and vice‑versa. By applying these maps to arbitrary subsets of merging blocks and successions they construct a bijection ψ_{R,S} that exchanges the statistics mb and suc. Consequently they obtain a q‑t symmetry: ∑{π∈Π₀ⁿ} q^{mb(π)} t^{suc(π)} = ∑{π∈Π₀ⁿ} t^{mb(π)} q^{suc(π)}. From this symmetry it follows that the distributions of merging blocks and of successions are identical.

The paper then studies the block‑generating polynomials for the three families: merging‑free partitions, separated partitions, and their intersection (partitions that are both merging‑free and separated). Using the recursions derived from the insertion of ±n into existing blocks, they prove that all these polynomials are real‑rooted, which implies unimodality and log‑concavity.

In Section 4 a bijection is presented that “flattens’’ a type B merging‑free partition into a Stirling permutation (a permutation of a multiset where each integer appears twice and the two copies are separated only by larger entries). This flattening preserves the block structure and prepares the ground for the study of permutation statistics.

Section 5 extends the flattening to signed permutations, defining the set RBₙ of signed permutations obtained from all type B merging‑free partitions. The main statistic examined is the descent number des(σ). The authors introduce a group action called valley‑hopping, originally used for Eulerian polynomials, which acts on RBₙ by moving elements across valleys while preserving the multiset of signs. They prove that the descent generating polynomial F_n(x)=∑{σ∈RBₙ} x^{des(σ)} is γ‑positive: there exist non‑negative integers γ₀,…,γ{⌊n/2⌋} such that F_n(x)=∑_{i=0}^{⌊n/2⌋} γ_i x^i (1+x)^{n−2i}. A combinatorial interpretation of the γ‑coefficients is given in terms of the number of valleys and peaks in the signed permutations. Moreover, the valley‑hopping action makes the descent statistic homomesic: the average descent number is the same on every orbit of the action.

The paper concludes by emphasizing that the identified families provide new combinatorial models counted by Dowling numbers, that their generating functions enjoy strong algebraic properties (real‑rootedness, γ‑positivity), and that the valley‑hopping framework yields both symmetry and homomesy results. The authors suggest further exploration of other statistics, q‑analogs, and extensions to other Coxeter groups.