  It is becoming increasingly clear that the supercharacter theory of the
finite group of unipotent upper-triangular matrices has a rich combinatorial
structure built on set-partitions that is analogous to the partition
combinatorics of the classical representation theory of the symmetric group.
This paper begins by exploring a connection to the ring of symmetric functions
in non-commuting variables that mirrors the symmetric group's relationship with
the ring of symmetric functions. It then also investigates some of the
representation theoretic structure constants arising from the restriction,
tensor products and superinduction of supercharacters in this context.
