  Recently, Babson and Steingrimsson have introduced generalised permutation
patterns that allow the requirement that two adjacent letters in a pattern must
be adjacent in the permutation. We consider pattern avoidance for such
patterns, and give a complete solution for the number of permutations avoiding
any single pattern of length three with exactly one adjacent pair of letters.
We also give some results for the number of permutations avoiding two different
patterns. Relations are exhibited to several well studied combinatorial
structures, such as set partitions, Dyck paths, Motzkin paths, and involutions.
Furthermore, a new class of set partitions, called monotone partitions, is
defined and shown to be in one-to-one correspondence with non-overlapping
partitions.
