A Bijectional Attack on the Razumov-Stroganov Conjecture

By Arvind Ayyer and Doron Zeilberger


Written: Dec. 2, 2008.

Exclusively published in the Personal Journal of Ekhad and Zeilberger and arxiv.org

The Razumov-Stroganov Conjecture is one of the most intriguing open problems in mathematics. Alas, its usual formulation is pretty intimidating, using the jargon of "advanced" mathematical physics. We realized that its statement is really elementary and easy-to-understand. This does not mean that our formulation (or rather our stronger statement) is easier-to-prove, but it very well might. Anyway, being kind and unselfish individuals, we followed the advice in Dr. Z.'s Opinion 39, and invite anyone to try to finish up our proof plan. Good luck!

Maple Packages

Important: This article is accompanied by Maple packages
  • RS, written by Arvind Ayyer.
  • FPL, written by Doron Zeilberger.

Sample Input and Output

To verify the Razumov-Stroganov conjecture for k from 1 to 7, the input yields the output.
Nice pictures of Fully Packed Loops can be found in this directory.
Added Dec. 3, 2008: Read Richard Stanley's interesting message.
Added May 6, 2010: Luigi Cantini and Andreas Sportiello beat us! They found a brilliant proof. Nevertheless, it is still possible that our suggested approach would lead to an even nicer proof.
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