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Concept
Stirling numbers of the second kind
Summary
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or . Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. They are named after James Stirling.
The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the second kind. Identities linking the two kinds appear in the article on Stirling numbers.
The Stirling numbers of the second kind, written or or with other notations, count the number of ways to partition a set of labelled objects into nonempty unlabelled subsets. Equivalently, they count the number of different equivalence relations with precisely equivalence classes that can be defined on an element set. In fact, there is a bijection between the set of partitions and the set of equivalence relations on a given set. Obviously,
for n ≥ 0, and for n ≥ 1,
as the only way to partition an n-element set into n parts is to put each element of the set into its own part, and the only way to partition a nonempty set into one part is to put all of the elements in the same part. Unlike Stirling numbers of the first kind, they can be calculated using a one-sum formula:
The Stirling numbers of the second kind may also be characterized as the numbers that arise when one expresses powers of an indeterminate x in terms of the falling factorials
(In particular, (x)0 = 1 because it is an empty product.)
In other words
Various notations have been used for Stirling numbers of the second kind. The brace notation was used by Imanuel Marx and Antonio Salmeri in 1962 for variants of these numbers.Antonio Salmeri, Introduzione alla teoria dei coefficienti fattoriali, Giornale di Matematiche di Battaglini 90 (1962), pp. 44–54.
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