A Very Short (Bijective!) Proof of Touchard's Catalan Identity
By
Amitai Regev, Nathaniel Shar, and Doron Zeilberger
.pdf
.ps
.tex
Posted: π Day, 2015
This version: March 24, 2015.
It is still fun to find bijective proofs of identities, even if they are shaloshable!
Added March 24, 2015: It turns out that our nice bijection was too nice not to have been
known before. See the insightful comments
by Dominique GouyoBeauchamp, Kyle Petersen, and Dennis Stanton.
Added Nov. 24, 2021: Nachum Dershowitz pointed out the follwing beautiful
article,
that is relevant to our note.
Maple Package

TOUCHARD,
A short Maple package, implementing the bijection (and its inverse), and checking, empirically,
that it is indeed a bijection!
An Input and Output file for the Maple package TOUCHARD
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page