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 a 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 , … {\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 x 1 α 1 x 2 α 2 ⋯ x k α k {\displaystyle x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{k}^{\alpha _{k}}} is equal to the coefficient of the monomial x i 1 α 1 x i 2 α 2 ⋯ x i k α k {\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
i 1 < i 2 < ⋯ < i k {\displaystyle i_{1}<i_{2}<\cdots <i_{k}} indexing the variables and any positive integer sequence ( α 1 , α 2 , … , α k ) {\displaystyle (\alpha _{1},\alpha _{2},\ldots ,\alpha _{k})} of exponents. 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 {\displaystyle S_{n}}
on a polynomial ring in n {\displaystyle n} variables x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} . One such action of S n {\displaystyle S_{n}} permutes variables, changing a polynomial p ( x 1 , … , x n ) {\displaystyle p(x_{1},\dots ,x_{n})} by iteratively swapping pairs ( x i , x i + 1 ) {\displaystyle (x_{i},x_{i+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 {\displaystyle S_{n}} conditionally permutes variables, changing a polynomial p ( x 1 , … , x n ) {\displaystyle p(x_{1},\ldots ,x_{n})}
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