Powers of 2 in Balanced Grid Colourings

2025/04/30 by Nikolai Beluhov, Beluhov, Nikolai · 1 citation
Engineering · Mathematics · #05A05 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2504.21451

openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B(m, n) be the number of ways to colour a 2m × 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 s₂(m)s₂(n), where s₂(x) denotes the number of 1s 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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• Powers of 2 in High-Dimensional Lattice Walks

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