Noncommutative symmetric function
In mathematics, the noncommutative symmetric functions form a Hopf algebra analogous to the Hopf algebra of symmetric functions. The Hopf algebra of noncommutative symmetric functions was introduced by Israel M. Gelfand, Daniel Krob, and Alain Lascoux et al. (1995). It is noncommutative but cocommutative graded Hopf algebra. It has the Hopf algebra of symmetric functions as a quotient, and is a subalgebra of the Hopf algebra of permutations, and is the graded dual of the Hopf algebra of quasisymmetric function.