  In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for
noncommutative symmetric functions which enables one to count words and
permutations with restrictions on the lengths of their increasing runs. We
generalize Gessel's theorem to allow for a much wider variety of restrictions
on increasing run lengths, and use it to complete the enumeration of
permutations with parity restrictions on peaks and valleys, and to give a
systematic method for obtaining generating functions for permutation statistics
that are expressible in terms of increasing runs. Our methods can also be used
to obtain analogous results for alternating runs in permutations.
