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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 theHamiltonian of the XYZ spin chain. By specializing the parameters in thepartition function of the 8VSOS model with DWBC and reflecting end, we find anexplicit combinatorial expression for $q_n(z)$ in terms of the partitionfunction of the three-color model with the same boundary conditions. Bazhanovand Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. Weprove 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, wefind some results on the number of states with a given number of faces of eachcolor, and we compute strict bounds for the possible number of faces of eachcolor.

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