# Who first noticed that Stirling numbers of the second kind count partitions?

When the Stirling numbers of the second kind were introduced by James Stirling in 1730, it was not combinatorially; rather, the numbers ${n \brace k}$ were defined via the polynomial identity $$x^n = \sum_{k=1}^n {n \brace k} x(x-1)\ldots (x-(k-1)). ~~(\star)$$ Most modern treatments of the Stirling numbers introduce ${n \brace k}$ combinatorially, as the number of ways of partitioning a set of size $n$ into $k$ non-empty blocks, and then (combinatorially) derive identities such as ($\star$).

While preparing for some upcoming talks on topics related to Stirling numbers, I realized that I have no idea who it was who first observed that the numbers ${n \brace k}$ defined by ($\star$) have a combinatorial interpretation.

The earliest reference I can find is in W. Stevens, Significance of Grouping, Annals of Eugenics volume 8 (1937), pages 57--69 (available at http://onlinelibrary.wiley.com/doi/10.1111/j.1469-1809.1937.tb02160.x/abstract). This seems to predate any mention of Stirling numbers of the second kind'' on MathSciNet.

I imagine that this is a question that has been thoroughly researched --- does anyone know of a reference?

• Mathematical Reviews (the name for MathSciNet before it became a website) started in 1940, so it's not a surprise that anything from 1937 will predate (not just seem to predate) the written reviews you'll find on MathSciNet. It's also not a surprise that you'd like to avoid citing a journal on eugenics from the late 1930s. Oct 25 '14 at 16:58

Niels Nielsen, who coined the name "Stirling number of the second kind", gives the partition counting interpretation in his 1904 book Handbuch der Theorie der Gammafunktion, page 70.

There is a discussion of this question in Section 4 of P. Stein, A brief history of enumeration, in Science and Computers (G.-C. Rota, ed.), Academic Press, pp. 169-206. According to Stein, "$\dots$ the earliest reference I have been able to find for this basic distribution problem is Whitworth's Choice and Chance (5th edition, 1901) $\dots$." Stein points out that Whitworth was using a definition of Stirling numbers of the second kind different from (but equivalent to) Stirling's and that "it is virtually certain that Whitworth did not recognize the identity of [the two definitions]."

• Thanks! I looked at Whitworth's book, and found that he discusses the problem already in the 2nd ed. (1870). On p. 173, he proves that "the number of ways in which $n$ different things can be distributed into $r$ indifferent parcels, with no blank lots, is $n!$ times the coefficient of $x^n$ in the expansion of $((e^x-1)^r)/r!$". A scan is available here. I can't find the 1st ed. (1867), but the problem is unlikely to appear there: in the preface, Whitworth says that the chapter in which the problem is presented is new to the 2nd ed.. Nov 1 '14 at 2:40
• @David. Interesting. Thanks for checking this. Nov 2 '14 at 14:14

Donald E. Knuth in the second part of his article Two Notes on Notation details some of the research he did on Stirling numbers and their history. He also introduces interesting notational conventions linking binomial coefficients and Stirling numbers:

$\binom{n+1}{k}$ = $\binom{n}{k}$ + $\binom{n}{k-1}$

$\genfrac{[}{]}{0pt}{}{n+1}{k}$ = $n\genfrac{[}{]}{0pt}{}{n}{k}$ + $\genfrac{[}{]}{0pt}{}{n}{k-1}$

$\genfrac{\{}{\}}{0pt}{}{n+1}{k}$ = $k\genfrac{\{}{\}}{0pt}{}{n}{k}$ + $\genfrac{\{}{\}}{0pt}{}{n}{k-1}$

where

$\genfrac{[}{]}{0pt}{}{n}{k}$ = the number of permutations of n objects having k cycles

$\genfrac{\{}{\}}{0pt}{}{n}{k}$ = the number of partitions of n objects into k nonempty subsets

A copy of the article is available at: https://arxiv.org/pdf/math/9205211.pdf

The answer is in the article at page 12 where Knuth mentions that : "Christian Kramp [28] proved near the end of the eighteenth century that ... "

[28] Christian Kramp, “Coefficient des allgemeinen Gliedes jeder willkührlichen Potenz eines Infinitinomiums; Verhalten zwischen Coefficienten der Gleichungen und Summen der Produkte und der Potenzen ihrer Wurzeln; Transformationen und Substitution der Reihen durcheinander”, in Der polynomische Lehrsatz, edited by Carl Friedrich Hindenburg (Leipzig, 1796), 91–122.

• Knuth's article is very nice but it doesn't seem to address David Galvin's question. Aug 19 '17 at 20:09