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

Inspired by Alexander Yong’s recent critique of the Hirsch Citation index, we give an empirical (yet very convincing!) reproof of the asymptotic normality of the Hirsch citation index (alias size of Durfee square) with respect to the uniform distribution on the “sample space” of integer partitions of n. This result was first proved rigorously (but with much greater effort!) by the humans Rodney Canfield, Sylvie Corteel, and Carla Savage. In particular, we confirm the CanfieldCorteel-Savage rigorous evaluation of the average: ( √ 6 log 2 π ) √ n+O(1) = 0.5404446 √ n+O(1), and estimate the variance, numerically, as 0.0811 √ n+O(1), and get estimates extremely close to those of the standard Normal Distribution for the first 12 standardized moments. We also observe that what Yong calls “Rodney Canfield’s concentration conjecture”, that asserts that most of the “mass” is close to the average, follows immediately from the Canfield-Corteel-Savage 1998 result, since the variance is proportional to the average (with a rather small proportionality constant, namely 0.0811/0.5404446, that is approximately 0.1501). All the results in this article were obtained via straightforward symboland numbercrunching, by the aid of a Maple package called HIRSCH, that is available free of charge from http://www.math.rutgers.edu/~zeilberg/tokhniot/HIRSCH . The Cinderella Story of the Size of the Durfee Square 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 ([H]), 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 article, [CCS], 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→∞), with precise asymptotics for the mean and variance, they did not dream that one day their result should be of interest to anyone who has ever published a paper. Alexander Yong’s Critique of the h-index In the latest issue of the Notices of the American Mathematical Society, Alexander Yong contributed ([Y]) a very insightful critique of the Hirsch citation index([H]). Yong mentioned [CCS], but apparently missed its full significance. In particular, what Yong calls “Canfield’s concentration conjecture” is an immediate consequence of the fact, proved in [CCS], that what now is called the h-index is asymptotically normal, and the fact, also proved there, that the asymptotic variance is proportional to the asymptotic average.