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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 and exponential generating functions in symbolic combinatorics ： ウィキペディア英語版 Stirling numbers and exponential generating functions in symbolic combinatorics The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used. It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style notation that is used for them. This article uses the coefficient extraction operator $()$ for formal power series, as well as the (labelled) operators $\backslash mathfrak$ (for cycles) and $\backslash mathfrak$ (for sets) on combinatorial classes, which are explained on the page for symbolic combinatorics. Given a combinatorial class, the cycle operator creates the class obtained by placing objects from the source class along a cycle of some length, where cyclical symmetries are taken into account, and the set operator creates the class obtained by placing objects from the source class in a set (symmetries from the symmetric group, i.e. an "unstructured bag".) The two combinatorial classes (shown without additional markers) are * permutations (for unsigned Stirling numbers of the first kind): ::$\backslash mathcal\; =\; \backslash mathfrak(\backslash mathfrak(\backslash mathcal)),$ and * set partitions into non-empty subsets (for Stirling numbers of the second kind): :: $\backslash mathcal\; =\; \backslash mathfrak(\backslash mathfrak\_(\backslash mathcal)),$ where $\backslash mathcal$ is the singleton class. Warning: The notation used here for the Stirling numbers is not that of the Wikipedia articles on Stirling numbers; square brackets denote the signed Stirling numbers here. ==Stirling numbers of the first kind== The unsigned Stirling numbers of the first kind count the number of permutations of () with ''k'' cycles. A permutation is a set of cycles, and hence the set $\backslash mathcal\backslash ,$ of permutations is given by :$\backslash mathcal\; =\; \backslash mathfrak(\backslash mathcal\; \backslash mathfrak(\backslash mathcal)),\; \backslash ,$ where the singleton $\backslash mathcal$ marks cycles. This decomposition is examined in some detail on the page on the statistics of random permutations. Translating to generating functions we obtain the mixed generating function of the unsigned Stirling numbers of the first kind: :$G(z,\; u)\; =\; \backslash exp\; \backslash left(\; u\; \backslash log\; \backslash frac\; \backslash right)\; =$ \left(\frac \right)^u = \sum_^\infty \sum_^n \left|\left(n \\ k \end\right )\right| u^k \, \frac. Now the signed Stirling numbers of the first kind are obtained from the unsigned ones through the relation :$\backslash left(n\; \backslash \backslash \; k\; \backslash end\backslash right)\; =$ (-1)^ \left|\left(n \\ k \end\right )\right|. Hence the generating function $H(z,\; u)$ of these numbers is :$H(z,\; u)\; =\; G(-z,\; -u)\; =$ \left(\frac \right)^ = (1+z)^u = \sum_^\infty \sum_^n \left(n \\ k \end\right ) u^k \, \frac. A variety of identities may be derived by manipulating this generating function: :$(1+z)^u\; =\; \backslash sum\_^\backslash infty\; z^n\; =$ \sum_^\infty \frac \sum_^n \left(n \\ k \end\right ) u^k = \sum_^\infty u^k \sum_^\infty \frac \left(n \\ k \end\right ) = e^. In particular, the order of summation may be exchanged, and derivatives taken, and then ''z'' or ''u'' may be fixed. 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 and exponential generating functions in symbolic combinatorics」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.