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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 Solid partition ： ウィキペディア英語版 Solid partition In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon.〔P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.〕 A solid partition of $n$ is a three-dimensional array, $n\_$, of non-negative integers (the indices $i,\backslash \; j,\backslash \; k\backslash geq\; 1$) such that :$\backslash sum\_\; n\_=n$ and :$$ n_ \leq n_\quad,\quad n_ \leq n_\quad\text\quad n_ \leq n_\quad,\quad \forall\ i,\ j \text k\ . Let $p\_3(n)$ denote the number of solid partitions of $n$. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.〔G. E. Andrews, ''The theory of partitions'', Cambridge University Press, 1998.〕 == Ferrers diagrams for solid partitions == Another representation for solid partitions is in the form of Ferrers diagrams. The Ferrers diagram of a solid partition of $n$ is a collection of $n$ points or ''nodes'', $\backslash lambda=(\backslash mathbf\_1,\backslash mathbf\_2,\backslash ldots,\backslash mathbf\_n)$, with $\backslash mathbf\_i\backslash in\; \backslash mathbb\_^4$ satisfying the condition:〔A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for'' ''m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097–1100.〕 :Condition FD: If the node $\backslash mathbf=(a\_1,a\_2,a\_3,\; a\_4)\backslash in\; \backslash lambda$, then so do all the nodes $\backslash mathbf=(y\_1,y\_2,y\_3,y\_4)$ with $0\backslash leq\; y\_i\backslash leq\; a\_i$ for all $i=1,2,3,4$. For instance, the Ferrers diagram :$$ \left( \begin 0\\ 0\\ 0 \\ 0 \end \begin 0\\ 0\\ 1 \\ 0 \end \begin 0\\ 1\\ 0 \\ 0 \end \begin1 \\ 0 \\ 0 \\ 0 \end \begin 1 \\ 1\\ 0 \\ 0 \end \right) \ , where each column is a node, represents a solid partition of $5$. There is a natural action of the permutation group $S\_4$ on a Ferrers diagram – this corresponds to permuting the four coordinates of all nodes. This generalises the operation denoted by conjugation on usual partitions. 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「Solid 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.