On multi-avoidance of generalized patterns

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 Sń, and 2E_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. More-over, 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.