On the proportion of derangements in affine classical groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914529287340032 |
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| author | Anzanello, Jessica |
| author_facet | Anzanello, Jessica |
| contents | We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $λ=(λ_1, \dots, λ_m)$ into $m$ parts, with $λ_1\ge \dots \ge λ_m$, such that either $λ_1=1$ or $λ_{k-1}>λ_k=k$ for some $k \in \{2, \dots,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07093 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the proportion of derangements in affine classical groups Anzanello, Jessica Combinatorics Group Theory 20B05, 05A17, 05A19 We derive exact formulas for the proportions of derangements and of derangements of $p$-power order in the affine classical groups $AU_m(q)$, $ASp_{2m}(q)$, $AO_{2m+1}(q)$ and $AO^{\pm}_{2m}(q)$, where $p$ denotes the characteristic of the defining finite field. In the unitary case, the formulas rely on a result on partitions of independent interest: we obtain a generating function for integer partitions $λ=(λ_1, \dots, λ_m)$ into $m$ parts, with $λ_1\ge \dots \ge λ_m$, such that either $λ_1=1$ or $λ_{k-1}>λ_k=k$ for some $k \in \{2, \dots,m\}$. In the symplectic and orthogonal cases, the proofs of the formulas reduce to verifying three $q$-polynomial identities conjectured by the author and later proved by Fulman and Stanton. |
| title | On the proportion of derangements in affine classical groups |
| topic | Combinatorics Group Theory 20B05, 05A17, 05A19 |
| url | https://arxiv.org/abs/2508.07093 |