  In recent work, Zeilberger and the author used a functional equations
approach for enumerating permutations with r occurrences of the pattern 12...k.
In particular, the approach yielded a polynomial-time enumeration algorithm for
any fixed nonnegative r. We extend that approach to patterns of the form
12...(k-2)(k)(k-1) by deriving analogous functional equations and using them to
develop similar algorithms that enumerate permutations with r occurrences of
the pattern. We also generalize those techniques to handle patterns of the form
23...k1 and derive analogous functional equations and enumeration algorithms.
Finally, we show how the functional equations and algorithms can be modified to
track inversions as well as handle multiple patterns simultaneously. This paper
is accompanied by Maple packages that implement the algorithms described.
