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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 Gaussian binomial coefficient ： ウィキペディア英語版 Gaussian binomial coefficient In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or ''q''-binomial coefficients) are ''q''-analogs of the binomial coefficients. ==Definition== The Gaussian binomial coefficients are defined by :$\_q$ = \begin \frac)} & r \le m \\ 0 & r>m \end where ''m'' and ''r'' are non-negative integers. For the value is 1 since numerator and denominator are both empty products. Although the formula in the first clause appears to involve a rational function, it actually designates a polynomial, because the division is exact in Z(). Note that the formula can be applied for , and gives 0 due to a factor in the numerator, in accordance with the second clause (for even larger ''r'' the factor 0 remains present in the numerator, but its further factors would involve negative powers of ''q'', whence explicitly stating the second clause is preferable). All of the factors in numerator and denominator are divisible by , with as quotient a ''q'' number: :$()\_q=\backslash frac=\backslash sum\_q^i=1+q+q^2+\backslash cdots+q^;$ dividing out these factors gives the equivalent formula :$\_q=\backslash frac\backslash quad(r\backslash leq\; m),$ which makes evident the fact that substituting into $\backslash tbinom\; mr\_q$ gives the ordinary binomial coefficient $\backslash tbinom\; mr.$ In terms of the ''q'' factorial $()\_q!=()\_q()\_q\backslash cdots()\_q$, the formula can be stated as :$\_q=\backslash frac\backslash quad(r\backslash leq\; m),$ a compact form (often given as only definition), which however hides the presence of many common factors in numerator and denominator. This form does make obvious the symmetry $\backslash tbinom\; mr\_q=\backslash tbinom\; m\_q$ for . Instead of these algebraic expressions, one can also give a combinatorial definition of Gaussian binomial coefficients. The ordinary binomial coefficient $\backslash tbinom\; mr$ counts the -combinations chosen from an -element set. If one takes those elements to be the different character positions in a word of length , then each -combination corresponds to a word of length using an alphabet of two letters, say with copies of the letter 1 (indicating the positions in the chosen combination) and letters 0 (for the remaining positions). To obtain from this model the Gaussian binomial coefficient $\backslash tbinom\; mr\_q$, it suffices to count each word with a factor , where is the number of "inversions" of the word: the number of pairs of positions for which the leftmost position of the pair holds a letter 1 and the rightmost position holds a letter 0 in the word. It can be shown that the polynomials so defined satisfy the Pascal identities given below, and therefore coincide with the polynomials given by the algebraic definitions. A visual way to view this definition is to associate to each word a path across a rectangular grid with sides of height and width , from the bottom left corner to the top right corner, taking a step right for each letter 0 and a step up for each letter 1. Then the number of inversions of the word equals the area of the part of the rectangle that is to the bottom-right of the path. Unlike the ordinary binomial coefficient, the Gaussian binomial coefficient has finite values for $m\backslash rightarrow\; \backslash infty$ (the limit being analytically meaningful for |''q''|<1): :$\_q\; =\; \backslash lim\_\; \_q\; =\; \backslash frac$ 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「Gaussian binomial coefficient」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.