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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 Young–Fibonacci lattice ： ウィキペディア英語版 Young–Fibonacci lattice In mathematics, the Young–Fibonacci graph and Young–Fibonacci lattice, named after Alfred Young and Leonardo Fibonacci, are two closely related structures involving sequences of the digits 1 and 2. Any digit sequence of this type can be assigned a ''rank'', the sum of its digits: for instance, the rank of 11212 is 1 + 1 + 2 + 1 + 2 = 7. As was already known in ancient India, the number of sequences with a given rank is a Fibonacci number. The Young–Fibonacci lattice is an infinite modular lattice having these digit sequences as its elements, compatible with this rank structure. The Young–Fibonacci graph is the graph of this lattice, and has a vertex for each digit sequence. The Young–Fibonacci graph and the Young–Fibonacci lattice were both initially studied in two papers by and . They are named after the closely related Young's lattice and after the Fibonacci number of their elements at any given rank. ==Digit sequences with a given rank== A digit sequence with rank may be formed either by adding the digit 2 to a sequence with rank , or by adding the digit 1 to a sequence with rank . If is the function that maps to the number of different digit sequences of that rank, therefore, satisfies the recurrence relation defining the Fibonacci numbers, but with slightly different initial conditions: (there is one rank-0 string, the empty string, and one rank-1 string, consisting of the single digit 1). These initial conditions cause the sequence of values of to be shifted by one position from the Fibonacci numbers: where denotes the th Fibonacci number. In the ancient Indian study of prosody, the Fibonacci numbers were used to count the number of different sequences of short and long syllables with a given total length; if the digit 1 corresponds to a short syllable, and the digit 2 corresponds to a long syllable, the rank of a digit sequence measures the total length of the corresponding sequence of syllables. See the Fibonacci number article for details. 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「Young–Fibonacci lattice」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.