  We study the inner product of Bethe states in the inhomogeneous periodic XXX
spin-1/2 chain of length L, which is given by the Slavnov determinant formula.
We show that the inner product of an on-shell M-magnon state with a generic
M-magnon state is given by the same expression as the inner product of a
2M-magnon state with a vacuum descendent. The second inner product is
proportional to the partition function of the six-vertex model on a rectangular
Lx2M grid, with partial domain-wall boundary conditions.
