Let k 2 be an integer, and let S be the set of all binary strings in which every block of zeroes has length less than k, and every block of ones has length at least k. Find an unambiguous expression for S, and use it to prove that the generating series for S weighted by length is given by S(x) = [(1 x k )(1 x + x k )]/(1 2x + x 2 x k+1 + x 2k) .