Powers of 2 in Balanced Grid Colourings

TL;DR

The paper investigates the 2-adic valuation of the number $B(m,n)$ of balanced colourings of a $2m\times 2n$ grid, testing Bhattacharya's conjecture that $v_2(B(m,n))=s_2(m)s_2(n)$. It proves two infinite families: when $m=2^k$, the exponent is $s_2(n)$, and when $m=2^k+1$, the exponent is $2s_2(n)$. The authors develop a synthesis of domino-decomposition techniques, constant-term generating-function methods, and a carry-graph algorithm to bound and compute the 2-adic valuation via binary digit sums, enabling tight results in these regimes. These results advance the understanding of the conjecture and furnish methodological tools that may extend to other parameter ranges and higher-dimensional analogues.

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.

Theorems & Definitions (9)

• proof : Proof of Theorem \ref{['2k']}
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