Editing Stirling number (section)

{{Short description|Mathematical sequences in combinatorics}}
{{Use American English|date = March 2019}}
In [[mathematics]], '''Stirling numbers''' arise in a variety of [[Analysis (mathematics)|analytic]] and [[combinatorics|combinatorial]] problems. They are named after [[James Stirling (mathematician)|James Stirling]], who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730).{{sfn|Mansour|Schork|2015|p=5}} They were rediscovered and given a combinatorial meaning by Masanobu Saka in his 1782 ''Sanpō-Gakkai'' ''(The Sea of Learning on Mathematics)''.{{sfn|Mansour|Schork|2015|p=4}}<ref>{{Cite book |title=Combinatorics: Ancient & Modern. |publisher=Oxford University Press |year=2013 |isbn=978-0-19-965659-2 |editor-last=Wilson, R., & Watkins, J. J. |pages=26}}</ref>

Two different sets of numbers bear this name: the [[Stirling numbers of the first kind]] and the [[Stirling numbers of the second kind]]. Additionally, [[Lah numbers]] are sometimes referred to as Stirling numbers of the third kind.  Each kind is detailed in its respective article, this one serving as a description of relations between them.

A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics.  Moreover, all three can be defined as the number of partitions of ''n'' elements into ''k'' non-empty subsets, where each subset is endowed with a certain kind of order (no order, cyclical, or linear).