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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 Euler's triangle ： ウィキペディア英語版 Eulerian number In combinatorics, the Eulerian number ''A''(''n'', ''m''), is the number of permutations of the numbers 1 to ''n'' in which exactly ''m'' elements are greater than the previous element (permutations with ''m'' "ascents"). They are the coefficients of the Eulerian polynomials: :$A\_(t)\; =\; \backslash sum\_^\; A(n,m)\backslash \; t^.$ The Eulerian polynomials are defined by the exponential generating function : $$ \sum_^ A_(t)\, \frac = \frac}. The Eulerian polynomials can be computed by the recurrence :$A\_(t)\; =\; 1,\backslash ,$ : $$ A_(t) = t\, (1-t)\, A_^(t) + A_(t)\, (1+(n-1)\, t),\quad n \ge 1. An equivalent way to write this definition is to set the Eulerian polynomials inductively by :$A\_(t)\; =\; 1,\backslash ,$ : $$ A_(t) = \sum_^ \binom\, A_(t)\, (t-1)^,\quad n \ge 1. Other notations for ''A''(''n'', ''m'') are ''E''(''n'', ''m'') and $\backslash scriptstyle\; \backslash left\backslash langle\; \backslash right\backslash rangle$. ==History== In 1755 Leonhard Euler investigated in his book ''Institutiones calculi differentialis'' polynomials ''α''1(''x'') = 1, ''α''2(''x'') = ''x'' + 1, ''α''3(''x'') = ''x''2 + 4''x'' + 1, etc. (see the facsimile). These polynomials are a shifted form of what are now called the Eulerian polynomials ''A''''n''(''x''). ==Basic properties== For a given value of ''n'' > 0, the index ''m'' in ''A''(''n'', ''m'') can take values from 0 to ''n'' − 1. For fixed ''n'' there is a single permutation which has 0 ascents; this is the falling permutation (''n'', ''n'' − 1, ''n'' − 2, ..., 1). There is also a single permutation which has ''n'' − 1 ascents; this is the rising permutation (1, 2, 3, ..., ''n''). Therefore ''A''(''n'', 0) and ''A''(''n'', ''n'' − 1) are 1 for all values of ''n''. Reversing a permutation with ''m'' ascents creates another permutation in which there are ''n'' − ''m'' − 1 ascents. Therefore ''A''(''n'', ''m'') = ''A''(''n'', ''n'' − ''m'' − 1). Values of ''A''(''n'', ''m'') can be calculated "by hand" for small values of ''n'' and ''m''. For example : For larger values of ''n'', ''A''(''n'', ''m'') can be calculated using the recursive formula :$A(n,m)=(n-m)A(n-1,m-1)\; +\; (m+1)A(n-1,m).$ For example :$A(4,1)=(4-1)A(3,0)\; +\; (1+1)A(3,1)=3\; \backslash times\; 1\; +\; 2\; \backslash times\; 4\; =\; 11.$ Values of ''A''(''n'', ''m'') for 0 ≤ ''n'' ≤ 9 are: : The above triangular array is called the Euler triangle or Euler's triangle, and it shares some common characteristics with Pascal's triangle. The sum of row ''n'' is the factorial ''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「Eulerian 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.