Substantial progress has been made in the past in design of
algorithms for the class of holonomic sequences, which has very nice
closure properties. Although large, this class still fails to include
several important and relatively simple sequences, encountered in
combinatorial enumeration, such as ordinary powers, Stirling numbers of
the first and second kind, and Eulerian numbers. So it seems necessary to
turn attention to ``subholonomic sequences'' which retain some, but not all of
the nice properties of holonomic ones. The hope is that this may help
us in tackling certain easy-to-state but difficult-to-solve combinatorial
problems, such as enumeration of lattice paths in restricted regions, and
enumeration of restricted permutations.