Free Novikov algebras and the Hopf algebra of decorated multi-indices

We propose a combinatorial formula for the coproduct in a Hopf algebra of decorated multi-indices that recently appeared in the literature, which can be briefly described as the graded dual of the enveloping algebra of the free Novikov algebra generated by the set of decorations. Similarly to what happens for the Hopf algebra of rooted forests, the formula can be written in terms of admissible cuts. We also prove a combinatorial formula for the extraction-contraction coproduct for undecorated multi-indices, in terms of a suitable notion of covering subforest.