Some statistics on permutations avoiding generalized patterns

In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their lengths. Here we tackle the problem of refining this enumeration by considering the statistics “first/last entry”. We give results for at least one of the mentioned statistics for every generalized patterns of type (1,2) or (2,1) as well as for some cases of permutations avoiding a pair of generalized patterns of the above types.