The Schützenberger involution over Dyck paths

We describe a map ΓΓ from the set of Dyck paths of given semilength to itself that is the analog of the Schützenberger involution on standard Young tableaux. Afterwards, we examine the behavior of ΓΓ with respect to Knuth’s correspondence between pairs of standard Young tableaux of the same shape with at most two rows and Dyck paths. Finally, we exploit the previous results to describe a bijection between the set of 321321-avoiding centrosymmetric permutations of even length and the set of 321321-avoiding involutions of the same length.