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To any poset or quasi-poset is attached a lattice polytope, whose Ehrhartpolynomial we study from a Hopf-algebraic point of view. We use for this twointeracting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopfalgebra morphism taking its values in Q[X], we deduce from the interactingbialgebras an algebraic proof of the duality principle, a generalization and anew proof of a result on B-series due to Whright and Zhao, using a monoid ofcharacters on quasi-posets, and a generalization of Faulhaber's formula. Wealso give non-commutative versions of these results: polynomials are replacedby packed words. We obtain in particular a non-commutative duality principle.

Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials