A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain

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.