Structural properties of Potts model partition functions and chromatic polynomials for lattice strips

The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width L_y and arbitrary length L_x has the form Z(G,q,v) = ∑_j=1^NZ,G,λ c_Z,G,j(λ_Z,G,j)^Lx, where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet (v=-1) is the chromatic polynomial P(G,q). Using coloring and transfer matrix methods, we give general formulas for C_X,G = ∑_j=1^NX,G,λ c_X,G,j for X = Z, P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient c_Z,G,j of degree d in q: c^(d) = U_2d(√q/2), where U_n(x) is the Chebyshev polynomial of the second kind, we determine the number of λ_Z,G,j's with coefficient c^(d) in Z(G,q,v) for these cyclic strips of width L_y to be n_Z(L_y,d)=(2d+1) (L_y+d+1)^-1(_Ly-d^2Ly) for 0 ≤ d ≤ L_y and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λ_Z,G,j is calculated to be N_Z,Ly,λ=(_Ly^2Ly). Results are also presented for the analogous numbers n_P(L_y,d) and N_P,Ly,λ for P(G,q). We find that n_P(L_y,0)=n_P(L_y-1,1)= M_Ly-1 (Motzkin number), n_Z(L_y,0)=C_Ly (the Catalan number), and give an exact expression for N_P,Ly,λ. Our results for N_Z,Ly,λ and N_P,Ly,λ apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations N_Z,Ly,λ=2N_DA,tri,Ly and N_P,Ly,λ=2N_DA,sq,Ly, where N_DA,Λ,n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths N_Z,Ly,λ∼L_y^-1/24^Ly and N_P,Ly,λ∼L_y^-1/23^Ly as L_y→∞. Some general geometric identities for Potts model partition functions are also presented.