Hamilton cycles for involutions of classical types
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916103458914304 |
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| author | Gutierres, Gonçalo Mamede, Ricardo Santos, José Luis |
| author_facet | Gutierres, Gonçalo Mamede, Ricardo Santos, José Luis |
| contents | Let ${\mathcal W}_n$ denote any of the three families of classical Weyl groups: the symmetric groups ${\mathcal S}_n$, the hyperoctahedral groups (signed permutation groups) ${\mathcal S}^B_n$, or the even-signed permutation groups ${\mathcal S}^D_n$. In this paper we give an uniform construction of a Hamilton cycle for the restriction to involutions on these three families of groups with respect to a inverse-closed connecting set of involutions. This Hamilton cycle is optimal with respect to the Hamming distance only for the symmetric group ${\mathcal S}_n$.
We also recall an optimal algorithm for a Gray code for type $B$ involutions. A modification of this algorithm would provide a Gray Code for type $D$ involutions with Hamming distance two, which would be optimal. We give such a construction for ${\mathcal S}^D_4$ and ${\mathcal S}^D_5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hamilton cycles for involutions of classical types Gutierres, Gonçalo Mamede, Ricardo Santos, José Luis Combinatorics Let ${\mathcal W}_n$ denote any of the three families of classical Weyl groups: the symmetric groups ${\mathcal S}_n$, the hyperoctahedral groups (signed permutation groups) ${\mathcal S}^B_n$, or the even-signed permutation groups ${\mathcal S}^D_n$. In this paper we give an uniform construction of a Hamilton cycle for the restriction to involutions on these three families of groups with respect to a inverse-closed connecting set of involutions. This Hamilton cycle is optimal with respect to the Hamming distance only for the symmetric group ${\mathcal S}_n$. We also recall an optimal algorithm for a Gray code for type $B$ involutions. A modification of this algorithm would provide a Gray Code for type $D$ involutions with Hamming distance two, which would be optimal. We give such a construction for ${\mathcal S}^D_4$ and ${\mathcal S}^D_5$. |
| title | Hamilton cycles for involutions of classical types |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.12839 |