Descents and flag major index on conjugacy classes of colored permutation groups without short cycles
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918105153798144 |
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| author | Liu, Kevin Yin, Mei |
| author_facet | Liu, Kevin Yin, Mei |
| contents | We consider the descent and flag major index statistics on the colored permutation groups, which are wreath products of the form $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We show that the $k$-th moments of these statistics on $\mathfrak{S}_{n,r}$ will coincide with the corresponding moments on all conjugacy classes without cycles of lengths $1,2,\ldots,2k$. Using this, we establish the asymptotic normality of the descent and flag major index statistics on conjugacy classes of $\mathfrak{S}_{n,r}$ with sufficiently long cycles. Our results generalize prior work of Fulman involving the descent and major index statistics on the symmetric group $\mathfrak{S}_n$. Our methods involve an intricate extension of Fulman's work on $\mathfrak{S}_n$ combined with the theory of the degree for a colored permutation statistic, as introduced by Campion Loth, Levet, Liu, Sundaram, and Yin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02990 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Descents and flag major index on conjugacy classes of colored permutation groups without short cycles Liu, Kevin Yin, Mei Combinatorics 05A05, 05E16, 60C05 We consider the descent and flag major index statistics on the colored permutation groups, which are wreath products of the form $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We show that the $k$-th moments of these statistics on $\mathfrak{S}_{n,r}$ will coincide with the corresponding moments on all conjugacy classes without cycles of lengths $1,2,\ldots,2k$. Using this, we establish the asymptotic normality of the descent and flag major index statistics on conjugacy classes of $\mathfrak{S}_{n,r}$ with sufficiently long cycles. Our results generalize prior work of Fulman involving the descent and major index statistics on the symmetric group $\mathfrak{S}_n$. Our methods involve an intricate extension of Fulman's work on $\mathfrak{S}_n$ combined with the theory of the degree for a colored permutation statistic, as introduced by Campion Loth, Levet, Liu, Sundaram, and Yin. |
| title | Descents and flag major index on conjugacy classes of colored permutation groups without short cycles |
| topic | Combinatorics 05A05, 05E16, 60C05 |
| url | https://arxiv.org/abs/2503.02990 |