  We study the connection between the three-color model and the polynomials
$q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the
Hamiltonian of the XYZ spin chain. By specializing the parameters in the
partition function of the 8VSOS model with DWBC and reflecting end, we find an
explicit combinatorial expression for $q_n(z)$ in terms of the partition
function of the three-color model with the same boundary conditions. Bazhanov
and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We
prove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$
have positive integer coefficients. Furthermore, for the three-color model, we
find some results on the number of states with a given number of faces of each
color, and we compute strict bounds for the possible number of faces of each
color.
