Abstract
We use the material of Parts I and II to obtain further results in sequence enumeration. These include the enumeration of sequences with respect to both maximal π1 and π2‐paths, and the enumeration of sequences with distinguished substrings. We use the material to derive an enumerative proof of a generalization of a result of Foata and Schützenberger. Finally, we reconsider the enumeration of permutations with periodic pattern and show that the required generating function may be exhibited as the solution of a set of first‐order nonlinear differential equations. Subsequent work has shown these to be matrix Riccati equations, although we refer to them here as Volterra equations.
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