Stirling Numbers

Stirling numbers arise in a variety of analytic and combinatorial problems and are named after James Stirling, who introduced them in a purely algebraic setting in his 1730 book Methodus differentialis; they were rediscovered and given a combinatorial meaning by Masanobu Saka in his 1782 work Sanpo-Gakkai. Two different sets of numbers carry the name, the Stirling numbers of the first kind and of the second kind, and Lah numbers are sometimes called Stirling numbers of the third kind. A property common to all three kinds is that they describe coefficients relating three different sequences of polynomials that arise often in combinatorics, and all three can be defined as counting the ways of partitioning a set of n elements into k nonempty subsets in which each subset carries a certain kind of order, either no order, cyclical order, or linear order.