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 2nβ1, the number of involutions in Snβ,
and 2Enβ, where Enβ 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 xβyβz (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.Comment: 26 page