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).

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 ${\displaystyle x_{1},x_{2},x_{3},\dots }$ with coefficients in R, which are shift invariant in the sense that the coefficient of the monomial ${\displaystyle x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{k}^{\alpha _{k}}}$ is equal to the coefficient of the monomial ${\displaystyle x_{i_{1}}^{\alpha _{1}}x_{i_{2}}^{\alpha _{2}}\cdots x_{i_{k}}^{\alpha _{k}}}$ for any strictly increasing sequence of positive integers ${\displaystyle i_{1}<i_{2}<\cdots <i_{k}}$ indexing the variables and any positive integer sequence ${\displaystyle (\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})}$ of exponents. Much of the study of quasisymmetric functions is based on that of symmetric functions.

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