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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 Major index ： ウィキペディア英語版 Major index In mathematics (and particularly in combinatorics), the major index of a permutation is the sum of the positions of the descents of the permutation. In symbols, the major index of the permutation ''w'' is : $$ \operatorname(w) = \sum_ i. For example, if ''w'' is given in one-line notation by ''w'' = 351624 (that is, ''w'' is the permutation of such that ''w''(1) = 3, ''w''(2) = 5, etc.) then ''w'' has descents at positions 2 (from 5 to 1) and 4 (from 6 to 2) and so maj(''w'') = 2 + 4 = 6. This statistic is named after Major Percy Alexander MacMahon who showed in 1913 that the distribution of the major index on all permutations of a fixed length is the same as the distribution of inversions. That is, the number of permutations of length ''n'' with ''k'' inversions is the same as the number of permutations of length ''n'' with major index equal to ''k''. (These numbers are known as ''Mahonian numbers'', also in honor of MacMahon.〔M. Bóna, Combinatorics of Permutations, 2004, p. 43ff, ISBN 1-58488-434-7.〕) In fact, a stronger result is true: the number of permutations of length ''n'' with major index ''k'' and ''i'' inversions is the same as the number of permutations of length ''n'' with major index ''i'' and ''k'' inversions, that is, the two statistics are equidistributed. For example, the number of permutations of length 4 with given major index and number of inversions is given in the table below. :$\backslash begin$ & 0&1&2&3&4&5&6 \\ \hline 0&1&0&0&0&0&0&0 \\ 1&0&1&1&1&0&0&0 \\ 2&0&1&2&1&1&0&0 \\ 3&0&1&1&2&1&1&0 \\ 4&0&0&1&1&2&1&0 \\ 5&0&0&0&1&1&1&0 \\ 6&0&0&0&0&0&0&1 \end == References == 〔 *. 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「Major index」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.