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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2601.17174 |
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| _version_ | 1866914294588768256 |
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| author | Alexandersson, Per Beyene, Fufa Mantaci, Roberto |
| author_facet | Alexandersson, Per Beyene, Fufa Mantaci, Roberto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17174 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some Families of Type $B$ Set Partitions Counted by the Dowling Numbers Alexandersson, Per Beyene, Fufa Mantaci, Roberto Combinatorics 05A05 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. |
| title | Some Families of Type $B$ Set Partitions Counted by the Dowling Numbers |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2601.17174 |