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

Barnabei, M.; Bonetti, F.; Silimbani, M.

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

Authors

Barnabei, M.

Bonetti, F.

Silimbani, M.

Description

We exploit Krattenthaler's bijection between the set $S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classical pattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distribution of every consecutive pattern of length 3 on the set $S_n(3\textrm{-}1\textrm{-}2)$. We show that these consecutive patterns split into 3 equidistribution classes, by means of an involution on Dyck paths due to E.Deutsch. In addition, we state equidistribution theorems concerning triplets of statistics relative to the occurrences of the consecutive patterns of length 3 in a permutation.
18 pages, 7 figures