Doron Gepner's Statistics on Words in {1,2,3}* is (Most Probably) Asymptotically Logistic

By Doron Zeilberger

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(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger)

Written: March 31, 2016

Dedicated to my friend and hero, Doron Gepner (b. March 31, 1956), on his 60th birthday
ד ו ר ו ן   ל ד ו ר ו ן   מ ד ו ר ו ן

I first met Doron Gepner in 1980, when he was a Physics graduate student at the Weizmann Institute of Science, and I was a young ח ו ק ר   ב כ י ר. Already then Doron was a legend, since he was the first person in Israel, as far as I know, to have solved Rubik's cube completely from scratch, using group-theoretical methods. I was so impressed that I asked him to present a guest-lecture in my graduate combinatorics class, and the students loved it.

Doron then went on to do seminal work in theoretical physics, that, unfortunately, is over my head. But the part that is really interesting to me is his current work, greatly generalizing the celebrated Rogers-Ramanujan identities, and giving lots of new insight. I am sure that this work will lead to many future gems.

In this modest "gift" (Doron in Hebrew) to Doron, I continue work inspired by Gepner way back in 1987.

Happy birthday, Doron, and keep up the good work!

GEPNER.txt.

## Sample Input and Output Files for the Maple package GEPNER.txt

• If you want to see the the first 10 Gepner polynomials of the first kind, i.e. the weight-enumerators according to Gepner's statistics on permutations of length n, that determines the number of three-lettered subwords whose reduction belongs to the set {132,213,321}
the input file yields the output file

• If you want to see the the first 12 Gepner polynomials of the second kind, i.e. the weight-enumerators according to Gepner's statistics on words defined on the set of words with n 1's, n 2's, and n 3's, that determines the number of three-lettered subwords that happen to belong to the set {132,213,321}
the input file yields the output file

• If you want to see the explicit (rigorously proved!) expressions for the average (easy!), variance (much harder, but still possible, for humans), and the third through the 12th moments about the mean
the input file yields the output file

• If you want to see the explicit (rigorously proved!) expressions for the average (easy!), variance (much harder, but still possible, for humans), and the third (very easy, since it equals 0) and the fourth moment about the mean (beyond the scope of humans)
the input file yields the output file

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger