  It is well-known that the set $\mathbf I_n$ of involutions of the symmetric
group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set
of $n$-permutations that avoid the two vincular patterns $\underline{123},$
$\underline{132}.$ We consider a bijection $\Gamma$ from the set $\mathbf S_n$
to the set of histoires de Laguerre, namely, bicolored Motzkin paths with
labelled steps, and study its properties when restricted to $\mathbf
S_n(1\underline{23},1\underline{32}).$ In particular, we show that the set
$\mathbf S_n(\underline{123},{132})$ of permutations that avoids the
consecutive pattern $\underline{123}$ and the classical pattern $132$
corresponds via $\Gamma$ to the set of Motzkin paths, while its image under $F$
is the set of restricted involutions $\mathbf I_n(3412).$ We exploit these
results to determine the joint distribution of the statistics des and inv over
  $\mathbf S_n(\underline{123},{132})$ and over $\mathbf I_n(3412).$
  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.

Combinatorics