A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage)

Shalosh B. Ekhad and Doron Zeilberger

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First Written: Oct. 31, 2014

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

Once upon a time there was an esoteric and specialized notion, called "size of the Durfee square", of interest to at most 100 specialists in the whole world. Then it was kissed by a prince called Jorge Hirsch, and became the famous (and to quite a few people, infamous) h-index, of interest to every scientist, and scholar, since it tells you how productive a scientist (or scholar) you are! When Rodney Canfield, Sylvie Corteel, and Carla Savage wrote their beautiful 1998 article proving, rigorously, by a very deep and intricate analysis, the asymptotic normality of the random variable "size of Durfee square" defined on integer-partitions of n (as n goes to infinity), with precise asymptotics for the mean and variance, they did not dream that one day their result should be of interest to everyone who has ever published a paper.

However Canfield et. al. had to work really hard to prove their deep result. Here we take an "empirical" shorcut, that proves the same thing much faster (modulo routine number- and symbol-crunching). More importantly, the empirical methodology should be useful in many other cases where rigorous proofs are either too hard, or not worth the effort!

Note: Our "empirical" approach is still based on purely mathematical data. Alexander Yong's wonderful critique, (that inspired the present note), is a masterpiece of (sociological) "meta-mathematics", and uses actual, real-world data! It is strongly recommended.

Added Nov. 13, 2014: watch the lecture (produced by Matthew Russell):

Maple Package

Some Input and Output files for the Maple package HIRSCH

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Doron Zeilberger's Home Page