Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips

partial abstract: 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 $Z(G,q,v)={\sum }_{j=1}^{N_{Z,G,\lambda }}{c}_{Z,G,j}({\lambda }_{Z,G,j}{)}^{L_x}$, 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 $C_{X,G}={\sum }_{j=1}^{N_{X,G,\lambda }}{c}_{X,G,j}$ for $X=Z,P$ on cyclic and M"obius 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 $q$: $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, where $U_n(x)$ is the Chebyshev polynomial of the second kind, we determine the number of ${\lambda }_{Z,G,j}$'s with coefficient $c^{(d)}$ in $Z(G,q,v)$ for these cyclic strips of width $L_y$ to be $n_Z(L_y,d)=(2d+1)(L_y+d+1)^{-1} {2L_y \choose L_y-d}$for$0 \le d \le L_y$ and zero otherwise. For both cyclic and M"obius strips of these lattices, the total number of distinct eigenvalues ${\lambda }_{Z,G,j}$ is calculated to be $N_{Z,L_y,\lambda }=\left(\genfrac{}{}{0px}{}{2L_y}{L_y}\right)$. We point out that $N_{Z,L_y,\lambda }=2N_{DA,tri,L_y}$ and $N_{P,L_y,\lambda }=2N_{DA,sq,L_y}$, where $N_{DA,\mathrm{\Lambda },n}$ denotes the number of directed lattice animals on the lattice $\mathrm{\Lambda }$.