Pairs of cointeracting bialgebras recently appears in the literature of combinatorial Hopf algebras, with examples based on formal series,
on trees (Calaque, Ebrahimi-Fard, Manchon), graphs (Manchon), posets...
We will give several results obtained on pairs of cointeracting bialgebras:
actions on the group of characters, antipode, morphisms to quasi-symmetric functions... and we will give applications to Ehrhart poylnomials and chromatic polynomials.