  A labeled set partition is a partition of a set of integers whose arcs are
labeled by nonzero elements of an abelian group $A$. Inspired by the action of
the linear characters of the unitriangular group on its supercharacters, we
define a group action of $A^n$ on the set of $A$-labeled partitions of an
$(n+1)$-set. By investigating the orbit decomposition of various families of
set partitions under this action, we derive new combinatorial proofs of Coker's
identity for the Narayana polynomial and its type B analogue, and establish a
number of other related identities. In return, we also prove some enumerative
results concerning Andr\'e and Neto's supercharacter theories of type B and D.
