Seminar Details

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 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.