  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 $N_{Z,L_y,\lambda}={2L_y \choose L_y}$.
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 $N_{DA,\Lambda,n}$ denotes the number
of directed lattice animals on the lattice $\Lambda$.
