In [BabStein] Babson and Steingr\'{\i}msson introduced generalized
permutation patterns that allow the requirement that two adjacent letters in a
pattern must be adjacent in the permutation. In [Kit1] Kitaev considered
simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no
internal dashes, that is, where the patterns correspond to contiguous subwords
in a permutation. There either an explicit or a recursive formula was given for
all but one case of simultaneous avoidance of more than two patterns.
In this paper we find the exponential generating function for the remaining
case. Also we consider permutations that avoid a pattern of the form x−yz or
xy−z and begin with one of the patterns 12...k, k(k−1)...1, 23...k1,
(k−1)(k−2)...1k or end with one of the patterns 12...k, k(k−1)...1,
1k(k−1)...2, k12...(k−1). For each of these cases we find either the
ordinary or exponential generating functions or a precise formula for the
number of such permutations. Besides we generalize some of the obtained results
as well as some of the results given in [Kit3]: we consider permutations
avoiding certain generalized 3-patterns and beginning (ending) with an
arbitrary pattern having either the greatest or the least letter as its
rightmost (leftmost) letter.Comment: 18 page
