A Very Short (Bijective!) Proof of Touchard's Catalan Identity

Amitai Regev, Nathaniel Shar, and Doron Zeilberger

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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 Gouyo-Beauchamp, 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

An Input and Output file for the Maple package TOUCHARD

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Doron Zeilberger's Home Page