I am Sorry, Richard Ehrenborg and Margie Readdy, About Your Two Conjectures, But One is FAMOUS, While the Other is FALSE

By Doron Zeilberger

Exclusive for the Personal Journal of Ekhad and Zeilberger

Written: July 2, 2002

It is always nice to have lunch with brilliant (and nice) people, but there are also some dangers involved. They may get you hooked on a beautiful conjecture, and distract you from your main project of proving RH or Goldbach. That's what happened to me on June 11, 2002, when I had lunch with Richard Ehrenborg, Margie Readdy, and their adorable one-year-old son, Theodore. Margie and Richard told me that they discovered, empirically, that the number of Down-Up Involutions of length 2k equals k!, for k=1,2,3,4,5, and conjectured that it holds in general.

After wasting one week trying to prove this, in vain, I started doubting it. It took some effort to program it, since the brute-force program runs out of time and memory for k=6. But a more clever recursive program (see the procdeure nDUI in the accompanying Maple package EHRENBORG), easily showed that the conjecture is false for k=6 (and beyond). Hence Richard and Margie were victims of the pernicious "Law of Small Numbers".

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IMPORTANT: This article is accompanied by the Maple package EHRENBORG, mentioned above.
Added Jan. 3, 2006: Read the very interesting message from Richard Stanley

Added March 20, 2006: The above result is now one small part of Richard Stanley's very interesting paper Alternating Permutations and Symmetric Functions

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