Coassociative magmatic bialgebras and the Fine numbers

There are many interpretations of Fine's sequence 1, 0, 1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, .... E.g., $F_n$ counts planar trees with a root of even out-degree and $n$ edges.

We consider vector spaces equipped with a binary operation and a binary cooperation. We don't suppose that one is a morphism for the other (the Hopf case), but we suppose that they satisfy the unital infinitesimal relation. Studying the structure of such generalized bialgebras, we obtain a new interpretation of Fine's sequence, and we identify their primitive parts as MagFine-algebras. In the terminology of J.L.Loday, the triple of operads (As, Mag, MagFine) is a good triple of operads.

This is joint work with Jean-Louis Loday and Maria Ronco.