翻訳と辞書 Words near each other ・ Lah ・ Lah Ab ・ Lah Bid ・ Lah Deraz-e Deli Khomsir ・ Lah Farakh ・ Lah Farakh-e Mushemi ・ Lah Frakh Melleh Namdaran ・ Lah Gavi Dalun ・ Lah Jarui ・ Lah Mari-ye Jowkar ・ Lah Meleh ・ Lah Meleh-ye Olya-ye Jowkar ・ Lah Meleh-ye Sofla-ye Jowkar ・ Lah Meleh-ye Vosta-ye Jowkar ・ Lah Mohammad Khani ・ Lah number ・ Lah Savz ・ Lah Var Hoseyn-e Olya ・ Lah, Yaba ・ Lah-Di-Dah ・ Lah-e Deraz ・ Lah-e Deraz, Borujen ・ Lah-Lah ・ Lah-Lah's Adventures ・ Lah-Lah’s Adventures ・ Lah-Toh County, Idaho Territory ・ Laha ・ Laha (town) ・ Laha language ・ Laha language (Indonesia) Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lah number ： ウィキペディア英語版 Lah number In mathematics, the Lah numbers, discovered by Ivo Lah in 1955,〔(John Riordan, ''Introduction to Combinatorial Analysis'' ), Princeton University Press (1958, reissue 1980) ISBN 978-0-691-02365-6 (reprinted again in 2002 by Dover Publications).〕 are coefficients expressing rising factorials in terms of falling factorials. Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of ''n'' elements can be partitioned into ''k'' nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers. Unsigned Lah numbers : :$L(n,k)\; =\; \backslash frac.$ Signed Lah numbers : :$L\text{'}(n,k)\; =\; (-1)^n\; \backslash frac.$ ''L''(''n'', 1) is always ''n''!; in the interpretation above, the only partition of into 1 set can have its set ordered in 6 ways: :, , , , or ''L''(3, 2) corresponds to the 6 partitions with two ordered parts: :, , , , or ''L''(''n'', ''n'') is always 1 since, e.g., partitioning into 3 non-empty subsets results in subsets of length 1. : Adapting the Karamata-Knuth notation for Stirling numbers, it has been proposed to use the following alternative notation for Lah numbers: :$L(n,k)=\backslash left\backslash lfloor\backslash begin\; n\; \backslash \backslash \; k\; \backslash end\backslash right\backslash rfloor.$ ==Rising and falling factorials== Let $x^$ represent the rising factorial $x(x+1)(x+2)\; \backslash cdots\; (x+n-1)$ and let $(x)\_n$ represent the falling factorial $x(x-1)(x-2)\; \backslash cdots\; (x-n+1)$. Then $x^\; =\; \backslash sum\_^n\; L(n,k)\; (x)\_k$ and $(x)\_n\; =\; \backslash sum\_^n\; (-1)^\; L(n,k)x^.$ For example, $x(x+1)(x+2)\; =\; x\; +\; x(x-1)\; +\; x(x-1)(x-2).$ Compare the third row of the table of values. 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「Lah number」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.