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We introduce new bases for the Hopf algebra of quasisymmetric functions thatrefine the symmetric powersum basis. These bases are expanded in terms ofquasisymmetric monomial functions by using fillings of matrices. We define theanalog of these bases in quasisymmetric functions of non-commuting variables.Our new bases have a (shifted) shuffle product and a deconcatenate coproduct.Finally, we describe a change of basis rule from the quasisymmetric powersumbasis to the quasisymmetric fundamental basis.

Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables