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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stirling numbers of the second kind ： ウィキペディア英語版 Stirling numbers of the second kind 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 $S(n,k)$ or $\backslash textstyle\; \backslash lbrace\backslash rbrace$.〔Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) ''Concrete Mathematics'', Addison–Wesley, Reading MA. ISBN 0-201-14236-8, p. 244.〕 Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. Stirling numbers of the second kind are one of two kinds of Stirling numbers, the other kind being called Stirling numbers of the first kind (or Stirling cycle numbers). Mutually inverse (finite or infinite) triangular matrices can be formed from the Stirling numbers of each kind according to the parameters ''n'', ''k''. ==Definition== The Stirling numbers of the second kind, written $S(n,k)$ or $\backslash lbrace\backslash textstyle\backslash rbrace$ or with other notations, count the number of ways to partition a set of $n$ labelled objects into $k$ nonempty unlabelled subsets. Given a $n$-element set $A$ and $\backslash pi\_:=\backslash$ where $\backslash pi\_A$ is the set of partitions of $A$, then $S(n,k):=\backslash mathrm(\backslash pi\_)$. Equivalently, they count the number of different equivalence relations with precisely $k$ equivalence classes that can be defined on an $n$ element set. In fact, there is a bijection between the set of partitions and the set of equivalence relations on a given set. Obviously, :$\backslash left\backslash \; =\; 1$ and for $n\; \backslash geq\; 1,\; \backslash left\backslash \; =\; 1.$ In fact, $\backslash pi\_=\backslash$. They can be calculated using the following explicit formula:〔 〕 :$\backslash left\backslash \; =\; \backslash frac\backslash sum\_^\; (-1)^\; \backslash binom\; j^n.$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stirling numbers of the second kind」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.