The Number of Rhombus Tilings of a Symmetric Hexagon which Contain a Fixed Rhombus on the Symmetry Axis, II

We compute the number of rhombus tilings of a hexagon with side lengths N, M,N , N, M, N, with N and M having the same parity, which contain a particular rhombus next to the center of the hexagon. The special case N = M of one of our results solves a problem posed by Propp. In the proofs, Hankel determinants featuring Bernoulli numbers play an important role.