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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 noncrossing partition ： ウィキペディア英語版 noncrossing partition In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The set of all noncrossing partitions is one of many sets enumerated by the Catalan numbers. The number of noncrossing partitions of an ''n''-element set with ''k'' blocks is found in the Narayana number triangle. ==Definition== A partition of a set ''S'' is a pairwise disjoint set of non-empty subsets, called "parts" or "blocks", whose union is all of ''S''. Consider a finite set that is linearly ordered, or (equivalently, for purposes of this definition) arranged in a cyclic order like the vertices of a regular ''n''-gon. No generality is lost by taking this set to be ''S'' = . A noncrossing partition of ''S'' is a partition in which no two blocks "cross" each other, i.e., if ''a'' and ''b'' belong to one block and ''x'' and ''y'' to another, they are not arranged in the order ''a x b y''. If one draws an arch based at ''a'' and ''b'', and another arch based at ''x'' and ''y'', then the two arches cross each other if the order is ''a x b y'' but not if it is ''a x y b'' or ''a b x y''. In the latter two orders the partition } is noncrossing. Equivalently, if we label the vertices of a regular ''n''-gon with the numbers 1 through ''n'', the convex hulls of different blocks of the partition are disjoint from each other, i.e., they also do not "cross" each other. The set of all non-crossing partitions of ''S'' are denoted $\backslash text(S)$. There is an obvious order isomorphism between $\backslash text(S\_1)$ and $\backslash text(S\_2)$for two finite sets $S\_1,S\_2$ with the same size. That is, $\backslash text(S)$ depends essentially only on the size of $S$ and we denote by $\backslash text(n)$ the non-crossing partitions on ''any'' set of size ''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「noncrossing 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.