Bijections from Dyck paths to 321-avoiding permutations revisited
There are at least three different bijections in the literature
from Dyck paths to 321-avoiding permutations, 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 ∘ L = K ∘ 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} ∘ B,
a measure of the difference between B and K, is the composition
of involutions L' ∘ L and turns out to be a very curious
bijection: as a permutation on Dyck n-paths it is an
n-th root of the "reverse path" involution. The proof of
this fact boils down to a geometric argument involving pairs of
nonintersecting lattice paths.
