  The $n^{th}$ partial flag incidence algebra of a poset $P$ is the set of
functions from $P^n$ to some ring which are zero on non-partial flag vectors.
These partial flag incidence algebras for $n>2$ are not commutative, not
unitary, and not associative. However, partial flag incidence algebras contain
generalized zeta, delta, and M\"obius functions which are finer and more
delicate invariants than their classical analogues. We also study some
generalized characteristic polynomials of posets which are not evaluations of
Tutte polynomials and compute them for Boolean lattices. Motivation for this
work came from studying the matroid Kazhdan-Lusztig polynomials where partial
flag Whitney numbers play a central role.
