On the number of partitions of the hypercube ${\bf Z}_q^n$ into large subcubes
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917881879461888 |
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| author | Tarannikov, Yuriy |
| author_facet | Tarannikov, Yuriy |
| contents | We prove that the number of partitions of the hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$. For the proof, we introduce the operation of the bang of a star matrix and demonstrate that any star matrix, except for a fractal, is expandable under some bang, whereas a fractal remains to be a fractal under any bang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04479 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the number of partitions of the hypercube ${\bf Z}_q^n$ into large subcubes Tarannikov, Yuriy Combinatorics Discrete Mathematics 05A18 We prove that the number of partitions of the hypercube ${\bf Z}_q^n$ into $q^m$ subcubes of dimension $n-m$ each for fixed $q$, $m$ and growing $n$ is asymptotically equal to $n^{(q^m-1)/(q-1)}$. For the proof, we introduce the operation of the bang of a star matrix and demonstrate that any star matrix, except for a fractal, is expandable under some bang, whereas a fractal remains to be a fractal under any bang. |
| title | On the number of partitions of the hypercube ${\bf Z}_q^n$ into large subcubes |
| topic | Combinatorics Discrete Mathematics 05A18 |
| url | https://arxiv.org/abs/2411.04479 |