Bijections from Dyck paths to 321-avoiding permutations revisited

There are at least three di erent bijections in the literature from Dyck paths to 321-avoiding permutations, due to Billey-Jockusch-Stanley, Krattenthaler, and Mansour-Deng-Du. How di erent are they? Denoting them B;K;M respectively, we show that M = B Æ L = K Æ L where L is the classical Kreweras-Lalanne involution on Dyck paths and L0, also an involution, is a sort of derivative of L. Thus K 1 Æ B, a measure of the di erence between B and K, is the product of involutions L ÆL 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.