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

Rings and Algebras