  It is well known that permutations avoiding any 3-length pattern are
enumerated by the Catalan numbers. If the three patterns 123, 132 and 213 are
avoided at the same time we obtain a class of permutations enumerated by the
Fibonacci numbers. We start from these permutations and make one or two
forbidden patterns disappear by suitably "generalizing" them. In such a way we
find several classes of permutations enumerated by integer sequences which lay
between the Fibonacci and Catalan numbers. For each class, we provide the
generating function according to the length of the permutations. Moreover, as a
result, we introduce a sort of "continuity" among the number sequences
enumerating these classes of permutations.
