  The notion of $S$-labeling, where $S$ is a subset of the symmetric group, is
a common generalization of signed $k$-coloring, signed $\mathbb{Z}_k$-coloring,
DP-coloring, group coloring, and coloring of gained graphs that was introduced
in 2019 by Jin, Wong, and Zhu. In this paper, we present a unified and simple
polynomial method for giving exponential lower bounds on the number of
colorings of an $S$-labeled graph. This algebraic technique allows us to prove
new lower bounds on the number of colorings of any $S$-labeling of graphs
satisfying certain sparsity conditions. This gives new lower bounds on the DP
color function, and consequently chromatic polynomial and list color function,
of families of planar graphs, and the number of colorings of signed graphs.
These bounds improve previously known results, or are the first such known
results.
