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.
From: Per Alexandersson [view email]
[v1]
Fri, 23 Jan 2026 21:08:47 UTC (27 KB)
[v2]
Fri, 30 Jan 2026 16:20:16 UTC (27 KB)
[v3]
Wed, 2 Sep 2026 09:37:31 UTC (28 KB)
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