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We exploit Krattenthaler's bijection between the set$S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classicalpattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distributionof every consecutive pattern of length 3 on the set$S_n(3\textrm{-}1\textrm{-}2)$. We show that these consecutive patterns splitinto 3 equidistribution classes, by means of an involution on Dyck paths due toE.Deutsch. In addition, we state equidistribution theorems concerning tripletsof statistics relative to the occurrences of the consecutive patterns of length3 in a permutation.

The distribution of consecutive patterns of length 3 in $3\textrm{-}1\textrm{-}2$-avoiding permutations