Permutations Restricted by Patterns of Type \((2,1)\)

Abstract

Recently, Babson and Steingrimsson (see \([BS]\)) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.

In this paper we study the generating functions for the number of permutations on \(n\) letters avoiding a generalized pattern \(ab-c\) where \((a,b,c) \in S_3\), and containing a prescribed number of occurrences of a generalized pattern \(cd-e\) where \((c,d,e) \in S_3\). As a consequence, we derive all the previously known results for this kind of problem, as well as many new results.