  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 $\mathcal{S}_n$,
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. 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.
