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 \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 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.
