  A natural generalization of single pattern avoidance is subset avoidance. A
complete study of subset avoidance for the case k=3 is carried out in [SS]. For
k>3 situation becomes more complicated, as the number of possible cases grows
rapidly. Recently, several authors have considered the case of general k when T
has some nice algebraic properties. Barcucci, Del Lungo, Pergola, and Pinzani
in [BDPP) treated the case when $T=T_1$ is the centralizer of k-1 and k under
the natural action of $S_k$ on [k]. Mansour and Vainshtein in [MVp] treated the
case when $T=T_2$ is maximal parabolic group of $S_k$.
  Recently, Babson and Steingrimsson (see [BS]) introduced generalized
permutations patterns that allow the requirement that two adjacent letters in a
pattern must be adjacent in the permutation.
  In this paper we present an analogue with generalization for the case $T_1$
and for the case $T_2$ by using generalized patterns instead of classical
patterns.

Combinatorics