On Multi-Avoidance of Generalized Patterns

Abstract

In [Kit1] Kitaev discussed simultaneous avoidance of two \(3\)-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such \(n\)-permutations are \(2^{n-1}\), the number of involutions in \(S_n\), and \(2^{E_n}\), where \(E_n\) is the \(n\)-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases.

To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form \(x-y-z\) (a classical \(3\)-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a \(3\)-pattern, begin with a certain pattern, and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized \(3\)-pattern and beginning and ending with increasing or decreasing patterns.