algebra of symmetric functions as the subalgebra of QSym spanned by the monomial symmetric functions and all formal power series where the sum is over all compositions which rearrange to the partition . Moreover, we have . For example, and Other important bases for quasisymmetric functions include the basis of quasisymmetric Schur functions, the "type I" and "type II" quasisymmetric power sums, and bases related to enumeration in matroids. Applications Quasisymmetric functions have been applied in enumerative combinatorics, symmetric function theory, representation theory, and number theory. Applications of quasisymmetric functions include enumeration of P-partitions, permutations, tableaux, chains of posets, reduced decompositions in finite Coxeter groups (via Stanley symmetric functions), and parking functions. In symmetric function theory and representation theory, applications include the study of Schubert polynomials, Macdonald polynomials, Hecke algebras, and Kazhdan–Lusztig polynomials. Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions. Related algebras As a graded Hopf algebra, the dual of the ring of quasisymmetric functions is the ring of noncommutative symmetric functions. Every symmetric function is also a quasisymmetric function, and hence the ring of symmetric functions is a subalgebra of the ring of quasisymmetric functions. The ring of quasisymmetric functions is the terminal object in category of graded Hopf algebras with a single character. Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions. One example of this is the peak algebra. Other related algebras The Malvenuto–Reutenauer algebra is a Hopf algebra based on permutations that relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions, (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram. The duality between QSym and NSym mentioned above is reflected in the main diagonal of this diagram. Many related Hopf algebras were constructed from Hopf monoids in the category of species by Aguiar and Majahan. One can also construct the ring of quasisymmetric functions in noncommuting variables. References External links BIRS Workshop on Quasisymmetric Functions Algebraic combinatorics Types of functions Polynomials Ring theory Hopf algebras