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We uncover the structure of the space of symmetric functions innon-commutative variables by showing that the underlined Hopf algebra is bothfree and co-free. We also introduce the Hopf algebra of quasi-symmetricfunctions in non-commutative variables and define the product and coproduct onthe monomial basis of this space and show that this Hopf algebra is free andcofree. In the process of looking for bases which generate the space we defineorders on the set partitions and set compositions which allow us to definebases which have simple and natural rules for the product of basis elements.Comment: 22 pages; 9 figures in eps for

THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE