Proof of a Conjecture of Philippe Di Francesco and Paul Zinn-Justin related to the qKZ equations and to Dave Robbins' Two Favorite Combinatorial Objects


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Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
First Written: March 21, 2007. This version: March 27, 2007. [incorporating remarks of referee Philippe Di Francesco].

This is a "homework assignment", assigned to me by the great mathematical physicist Philippe Di Francesco. It was mentioned in Philippe's plenary talk at the CRM, Montreal, conference on the combinatorics of Statistical Physics (wonderfully organized by Pierre Leroux) that took place a month ago. When I saw the conjecture, I was sure that I'll be able to prove it by the next day, and announce, and possibly present it, in my own plenary talk. Well, if it would have been that simple, Philippe would have done it himself. In fact, it took me exactly four weeks. Of course, I also worked on my other projects, but it took me many more hours than I would like to admit.

Added March 27, 2007: This paper was carefully refereed by Philippe Di Francesco. See his referee report. I agree with all his suggestions, and have made them in this revised version.
Added March 25, 2007: Christian Krattenthaler has just informed me that he found another proof. Once it is available, I will put a link to it here.

Added March 29, 2007: Apparently, Alain Lascoux did it as well, using "fancy" methods (but it may be just a matter of language). Read his intriguing message

Added April 2, 2007: Hjalamar Rosengren independetly proved the Di Francesco-Zinn-Justin conjecture, using a similar approach, but at the end using partial fractions rather than Lagrange Interpolation.

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