  We present exact results on the partition function of the $q$-state Potts
model on various families of graphs $G$ in a generalized external magnetic
field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of
the total set of possible spin values, $Z(G,q,s,v,w)$, where $v$ and $w$ are
temperature- and field-dependent Boltzmann variables. We remark on differences
in thermodynamic behavior between our model with a generalized external
magnetic field and the Potts model with a conventional magnetic field that
favors or disfavors a single spin value. Exact results are also given for the
interesting special case of the zero-temperature Potts antiferromagnet,
corresponding to a set-weighted chromatic polynomial $Ph(G,q,s,w)$ that counts
the number of colorings of the vertices of $G$ subject to the condition that
colors of adjacent vertices are different, with a weighting $w$ that favors or
disfavors colors in the interval $I_s$. We derive powerful new upper and lower
bounds on $Z(G,q,s,v,w)$ for the ferromagnetic case in terms of zero-field
Potts partition functions with certain transformed arguments. We also prove
general inequalities for $Z(G,q,s,v,w)$ on different families of tree graphs.
As part of our analysis, we elucidate how the field-dependent Potts partition
function and weighted-set chromatic polynomial distinguish, respectively,
between Tutte-equivalent and chromatically equivalent pairs of graphs.
