Hopf algebras in the free noncommutative cocommutative graded connected category exist with given dimension sequence iff the INVERTi transform is nonnegative.
The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables.Advances in Applied Mathematics, 47(3):536–544
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When are Hopf algebras determined by integer sequences?
Hopf algebras in the free noncommutative cocommutative graded connected category exist with given dimension sequence iff the INVERTi transform is nonnegative.