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associahedron
In mathematics, an associahedron ''K''''n'' is an (''n'' − 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of ''n'' letters and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with ''n'' + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s〔. Revised from a 1961 Ph.D. thesis, Princeton University, .〕 after earlier work on them by Dov Tamari.〔.〕
==Examples==
The one-dimensional associahedron ''K''3 represents the two parenthesizations ((''xy'')''z'') and (''x''(''yz'')) of three symbols, or the two triangulations of a square.
The two-dimensional associahedron represents the five parenthesizations of four symbols, or the five triangulations of a regular pentagon. It is itself a pentagon.
The three-dimensional associahedron ''K''5 is an enneahedron with nine faces and fourteen vertices, and its dual is the triaugmented triangular prism.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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