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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 plane partition ： ウィキペディア英語版 plane partition In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers $n\_$ (with positive integer indices ''i'' and ''j'') that is nonincreasing in both indices, that is, that satisfies :$n\_\; \backslash ge\; n\_\; \backslash quad\backslash mbox\backslash quad\; n\_\; \backslash ge\; n\_\; \backslash ,$ for all ''i'' and ''j'', and for which only finitely many of the ''n''''i'',''j'' are nonzero. A plane partitions may be represented visually by the placement of a stack of $n\_$ unit cubes above the point (''i'',''j'') in the plane, giving a three-dimensional solid like the one shown at right. The ''sum'' of a plane partition is :$n=\backslash sum\_\; n\_\; \backslash ,$ and PL(''n'') denotes the number of plane partitions with sum ''n''. For example, there are six plane partitions with sum 3: : \qquad \begin 1 & 1 \\ 1 & \end \qquad \begin 1 \\ 1 \\ 1 & \end \qquad \begin 2 & 1 & \end \qquad \begin 2 \\ 1 & \end \qquad \begin 3 \end so PL(3) = 6. (Here the plane partitions are drawn using matrix indexing for the coordinates and the entries equal to 0 are suppressed for readability.) == Ferrers diagrams for plane partitions == Another representation for plane partitions is in the form of Ferrers diagrams. The Ferrers diagram of a plane partition of $n$ is a collection of $n$ points or ''nodes'', $\backslash lambda=(\backslash mathbf\_1,\backslash mathbf\_2,\backslash ldots,\backslash mathbf\_n)$, with $\backslash mathbf\_i\backslash in\; \backslash mathbb\_^$ satisfying the condition:〔A. O. L. Atkin, P. Bratley, I. G. Macdonald and J. K. S. McKay, Some computations for ''m''-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.〕 :Condition FD: If the node $\backslash mathbf=(a\_1,a\_2,a\_3)\backslash in\; \backslash lambda$, then so do all the nodes $\backslash mathbf=(y\_1,y\_2,y\_3)$ with $0\backslash leq\; y\_i\backslash leq\; a\_i$ for all $i=1,2,3$. Replacing every node of a plane partition by a unit cube with edges aligned with the axes leads to the ''stack of cubes'' representation for the plane partition. 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「plane partition」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.