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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.21451 |
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| _version_ | 1866918005334605824 |
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| author | Beluhov, Nikolai |
| author_facet | Beluhov, Nikolai |
| contents | Let $B(m, n)$ be the number of ways to colour a $2m \times 2n$ grid in black and white so that, in each row and each column, half of the cells are white and half are black. Bhattacharya conjectured that the exponent of $2$ in the prime factorisation of $B(m, n)$ equals $s_2(m)s_2(n)$, where $s_2(x)$ denotes the number of $1$s in the binary expansion of $x$. We confirm this conjecture in some infinite families of special cases; most significantly, when $m$ is of the form either $2^k$ or $2^k + 1$ and $n$ is arbitrary. The proof when $m = 2^k + 1$ is substantially more difficult, and in connection with it we develop some general techniques for the analysis of inequalities between binary digit sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Powers of 2 in Balanced Grid Colourings Beluhov, Nikolai Combinatorics 05A05 Let $B(m, n)$ be the number of ways to colour a $2m \times 2n$ grid in black and white so that, in each row and each column, half of the cells are white and half are black. Bhattacharya conjectured that the exponent of $2$ in the prime factorisation of $B(m, n)$ equals $s_2(m)s_2(n)$, where $s_2(x)$ denotes the number of $1$s in the binary expansion of $x$. We confirm this conjecture in some infinite families of special cases; most significantly, when $m$ is of the form either $2^k$ or $2^k + 1$ and $n$ is arbitrary. The proof when $m = 2^k + 1$ is substantially more difficult, and in connection with it we develop some general techniques for the analysis of inequalities between binary digit sums. |
| title | Powers of 2 in Balanced Grid Colourings |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2504.21451 |