  Let G be a group of automorphisms of a ranked poset Q and let N_{k} denote
the number of orbits on the elements of rank k in Q. What can be said about the
N_{k} for standard posets, such as finite projective spaces or the Boolean
lattice? We discuss the connection of this question to the representation
theory of the group, and in particular to the inequalities of
Livingstone-Wagner and Stanley. We show that these are special cases of more
general inequalities which depend on the prime divisors of the group order. The
new inequalities often yield stronger bounds depending on the order of the
group.
