Bijections from Dyck paths to 321-avoiding permutations revisited

Abstract

There are (at least) three bijections from Dyck paths to 321-avoiding permutations in the literature, due to Billey-Jockusch-Stanley, Krattenthaler, and Mansour-Deng-Du. How different are they? Denoting them B,K,M respectively, we show that M = B \circ L = K \circ L' where L is the classical Kreweras-Lalanne involution on Dyck paths and L', also an involution, is a sort of derivative of L. Thus K^{-1} \circ B, a measure of the difference between B and K, is the product of involutions L' \circ L and turns out to be a very curious bijection: as a permutation on Dyck n-paths it is an nth root of the "reverse path" involution. The proof of this fact boils down to a geometric argument involving pairs of nonintersecting lattice paths.