  We calculate the partition function of the $q$-state Potts model on
arbitrary-length cyclic ladder graphs of the square and triangular lattices,
with a generalized external magnetic field that favors or disfavors a subset of
spin values $\{1,...,s\}$ with $s \le q$. For the case of antiferromagnet
spin-spin coupling, these provide exactly solved models that exhibit an onset
of frustration and competing interactions in the context of a novel type of
tensor-product $S_s \otimes S_{q-s}$ global symmetry, where $S_s$ is the
permutation group on $s$ objects.
