Hopf algebras in the free noncommutative cocommutative graded connected category exist with given dimension sequence iff the INVERTi transform is nonnegative.
Commutative combinatorial Hopf algebras.Journal of Algebraic Combinatorics, 28:65–95
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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.