  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. We consider n-permutations that avoid the
generalized pattern 1-32 and whose k rightmost letters form an increasing
subword. The number of such permutations is a linear combination of Bell
numbers. We find a bijection between these permutations and all partitions of
an $(n-1)$-element set with one subset marked that satisfy certain additional
conditions. Also we find the e.g.f. for the number of permutations that avoid a
generalized 3-pattern with no dashes and whose k leftmost or k rightmost
letters form either an increasing or decreasing subword. Moreover, we find a
bijection between n-permutations that avoid the pattern 132 and begin with the
pattern 12 and increasing rooted trimmed trees with n+1 nodes.
