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arXiv · 2504.21451

Powers of 2 in Balanced Grid Colourings

Nikolai Beluhov
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Abstract

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.

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Nikolai Beluhov. 2025-04-30. Powers of 2 in Balanced Grid Colourings. https://arxiv.org/abs/2504.21451

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