Powers of 2 in balanced grid colourings

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 s 2 (m)s 2 (n), where s 2 (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.