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In recent work, Zeilberger and the author used a functional equationsapproach for enumerating permutations with r occurrences of the pattern 12...k.In particular, the approach yielded a polynomial-time enumeration algorithm forany fixed nonnegative r. We extend that approach to patterns of the form12...(k-2)(k)(k-1) by deriving analogous functional equations and using them todevelop similar algorithms that enumerate permutations with r occurrences ofthe pattern. We also generalize those techniques to handle patterns of the form23...k1 and derive analogous functional equations and enumeration algorithms.Finally, we show how the functional equations and algorithms can be modified totrack inversions as well as handle multiple patterns simultaneously. This paperis accompanied by Maple packages that implement the algorithms described.

Approaches for enumerating permutations with a prescribed number of occurrences of patterns