Consecutive patterns in restricted permutations and involutions

It is well-known that the set of involutions of the symmetric group corresponds bijectively - by the Foata map - to the set of -permutations that avoid the two vincular patterns 123, 132. We consider a bijection Γ from the set to the set of histoires de Laguerre, namely, bicolored Motzkin paths with labelled steps, and study its properties when restricted to (123,132). In particular, we show that the set (123,132) of permutations that avoids the consecutive pattern 123 and the classical pattern 132 corresponds via Γ to the set of Motzkin paths, while its image under is the set of restricted involutions (3412). We exploit these results to determine the joint distribution of the statistics des and inv over (123,132) and over (3412). Moreover, we determine the distribution in these two sets of every consecutive pattern of length three. To this aim, we use a modified version of the well-known Goulden-Jacson cluster method.