Hopf algebras in the free noncommutative cocommutative graded connected category exist with given dimension sequence iff the INVERTi transform is nonnegative.
Grothendieck bialgebras, partition lattices, and symmetric functions in noncommutative variables.The Electronic Journal of Combinatorics, pages R75–R75
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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.