Note that this automorphism cannot be produced just by giving A127288 (the inverse of A127287) to function "tau" given in A127291. Instead, we have to use another algorithm given in A127300 and then conjugate it by A057164.

REFERENCES

Emeric Deutsch and Sergi Elizalde, A simple and unusual bijection for Dyck paths and its consequences, Annals of Combinatorics, 7 (2003), no. 3, 281-297.