  It is shown that the transfer matrix of the inhomogeneous nineteen-vertex
model with certain diagonal twisted boundary conditions possesses a simple
eigenvalue. This is achieved through the identification of a simple and
completely explicit solution of its Bethe equations. The corresponding
eigenvector is computed by means of the algebraic Bethe ansatz, and both a
simple component and its square norm are expressed in terms of the
Izergin-Korepin determinant. In the homogeneous limit, the vector coincides
with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown
that in a natural polynomial normalisation scheme its square norm and the
simple component coincide with generating functions for weighted enumeration of
alternating sign matrices.
