Actions and Identities on Set Partitions

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 An 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 André and Neto’s supercharacter theories of type B and D.