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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 Quasisymmetric function ： ウィキペディア英語版 Quasisymmetric function In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables. This ring generalizes the ring of symmetric functions. This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in ''n'' variables, as ''n'' goes to infinity. This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number ''n'' of variables (but its elements are neither polynomials nor functions). == Definitions == The ring of quasisymmetric functions, denoted QSym, can be defined over any commutative ring ''R'' such as the integers. Quasisymmetric functions are power series of bounded degree in variables $x\_1,x\_2,x\_3,\; \backslash dots$ with coefficients in ''R'', which are shift invariant in the sense that the coefficient of the monomial $x\_1^x\_2^\; \backslash cdots\; x\_k^$ is equal to the coefficient of the monomial $x\_^\; x\_^\backslash cdots\; x\_^$ for any strictly increasing sequence of positive integers indexing the variables and any positive integer sequence $(\backslash alpha\_1,\; \backslash alpha\_2,\backslash ldots,\backslash alpha\_k)$ of exponents.〔 Stanley, Richard P. ''Enumerative Combinatorics'', Vol. 2, Cambridge University Press, 1999. ISBN 0-521-56069-1 (hardback) ISBN 0-521-78987-7 (paperback).〕 Much of the study of quasisymmetric functions is based on that of symmetric functions. A quasisymmetric function in finitely many variables is a ''quasisymmetric polynomial''. Both symmetric and quasisymmetric polynomials may be characterized in terms of actions of the symmetric group $S\_n^,\backslash dots,\; x\_n$. One such action of $S\_n$ permutes variables, changing a polynomial $p(x\_1^)$ of variables having consecutive indices. Those polynomials unchanged by all such swaps form the subring of symmetric polynomials. A second action of $S\_n$ conditionally permutes variables, changing a polynomial $p(x\_1,\backslash ldots,x\_n)$ by swapping pairs $(x\_i^$ x_1^2 x_2 x_3 + x_1^2 x_2 x_4 + x_1^2 x_3 x_4 + x_2^2 x_3 x_4 + x_1 x_2^2 x_3 + x_1 x_2^2 x_4 + x_1 x_3^2 x_4 + x_2 x_3^2 x_4 \\ {} + x_1 x_2 x_3^2 + x_1 x_2 x_4^2 + x_1 x_3 x_4^2 + x_2 x_3 x_4^2. \, \end{align} 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「Quasisymmetric function」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.