Sketch of a Proof of an Intriguing Conjecture of Karola Mészáros and Alejandro Morales Regarding the Volume of the Dn Analog of the Chan-Robbins-Yuen Polytope
(Or: The Morris-Selberg Constant Term Identity Strikes Again!)

By Doron Zeilberger

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Written: July 10, 2014

[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and]

Dedicated to Dick Askey (b. June 4, 1933), from a 43-years-old to a 34-years-old , and thanks for preaching the importance of Constant Term Identities!

Added July 14, 2014: Jang Soo Kim independently found a proof, that is complete (not just a sketch), with all the steps rigorously justified!

In 1998, Clara S. Chan, David P. Robbins, and David S. Yuen conjectured and I proved, using the amazing Morris Constant Term Identity (that may be viewed as a contour-integral analog of the Selberg integral).

Recently Karola Mészáros and Alejandro Morales, conjectured a Dn analog of the Chan-Robbins-Yuen formula, getting another constant-term expression. This conjecture was also mentioned by Alejandro Morales in the Open Problems session that took place on June 23, 2014, ca. 4:50-5:00pm at the Stanley@70 conference [see his gorgeous slides].

It turns out that the Mészáros-Morales conjecture is a special case of a much more general constant term identity, and the proof in the present article is a "cheating" proof for two reasons:

but I am sure that many analysts can easily `fix' it.

Let me also comment that for any fixed dimension n, the new constant term identity is doable by the WZ method, and I am also sure that one can get a completely elementary WZ-style proof for general dimensions, by looking at the output for small n, and finding a pattern, like it was done for the original Selberg integral in section 6.5 of this masterpiece.

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