Applying the Cluster Method to Count Occurrences of Generalized Permutation Patterns

TL;DR

This paper extends the cluster method to count permutations with specific generalized patterns, providing recurrences and Maple implementations for efficient enumeration.

Contribution

It introduces a novel application of the cluster method to generalized permutation patterns with recurrences and computational tools.

Findings

Derived recurrences for counting permutations with generalized patterns

Implemented Maple procedures for practical computation

Simplified counting in the case of permutations

Abstract

We apply ideas from the cluster method to q-count the permutations of a multiset according to the number of occurrences of certain generalized patterns, as defined by Babson and Steingrimsson. In particular, we consider those patterns with three letters and one internal dash, as well as permutation statistics composed of counting the number of occurrences of multisets of such patterns. Counting is done via recurrences which simplify in the case of permutations. A collection of Maple procedures implementing these recurrences accompanies the article.