Automatic Enumeration of Generalized Ménage Numbers

Doron Zeilberger

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[Apperead in the Séminaire Lotharingien de Combinatoire, B71a (2014)]

First Written: Jan. 5, 2014.
This version: Feb. 3, 2014.

In fond memory of Alain Lascoux (1944-2013), one of the most CREATIVE and ORIGINAL and INTERESTING mathematicians that I have ever known

In Sept. 15-18, I gave three talks at the 71st session of the famous Séminaire Lotharingien de Combinatoire, on "Sieve Methods in Number Theory and Combinatorics". Since I am famously against laptop talks, I had to use the tiny whiteboard in the lecture room (in the magnificient former Bishop's castle of Bertinoro). But the first day was so beautiful, that I decided to take my "class" outside, and the whole talk was entirely oral, telling the history of sieve methods in number theory, from Eratosthenes, via Viggo Brun, all the way to the wonderful Cinderalla story of Yitang Zhang.

The two other talks were indoors. At the second talk I mentioned and briefly sketched, the Brydges-Spencer Lace Expansion, that lead, way back in the early-1990s, to the seminal work of Takashi Hara and Gordon Slade about the asymptotic behavior of the enumerating sequence of self-avoiding walks in dimensions five and up. The last talk, that formed the basis of the present article, was about counting restricted permutations via rook polynomials, and how computers can be taught to generate, in a few seconds, deep theorems, that took such great minds like Arthur Cayley, F.R.S., Sir Thomas Muir, Monsieur le colonel Charles Moreau (a decorated soldier, brilliant amateur mathematician, but not quite as good chess player), notable politician Charles-Ange Laisant, the great Major Percy MacMahon, Jacques Touchard, John Riordan (the master of ars combinatorica), the great algebraist Irving Kaplansky, the great enumerator Earl Glenn Whitehad, and numerous others (including algebaric combinatorics guru Richard Stanley, that covered rook polynomials in his classic EC1, but regretfully did not even mention John Riordan, his analog a generation earlier).

The venue was especially auspicious, as Bertinoro is the birthplace of one of the greatest medieval rabbis, Obadiah Bertinura, the great commentator of the Mishna. My good friend Omar Foda, who participated in the conference, found his house and kindly took a picture of me standing in front Obadiah's house, and another picture of my wife Jane and I standing in front of it

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