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

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]."

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.

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.