A set-operad of formal fractions and dendriform-like sub-operads

Frédéric Chapoton (1) , Florent Hivert (2) , Jean-Christophe Novelli (3)

1 ICJ - Institut Camille Jordan
2 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)
3 LIGM - Laboratoire d'Informatique Gaspard-Monge

We introduce an operad of formal fractions, abstracted from the Mould operads and containing both the Dendriform and the Tridendriform operads. We consider the smallest set-operad contained in this operad and containing four specific elements of arity two, corresponding to the generators and the associative elements of the Dendriform and Tridendriform operads. We obtain a presentation of this operad (by binary generators and quadratic relations) and an explicit combinatorial description using a new kind of bi-colored trees. Similar results are also presented for related symmetric operads.

Keywords: Combinatorics; dendriforms algebras; operads

Domains: Mathematics [math]; Combinatorics [math.CO]

Frédéric Chapoton, Florent Hivert, Jean-Christophe Novelli. A set-operad of formal fractions and dendriform-like sub-operads. Journal of Algebra, 2016, 465, pp.322-355. ⟨10.1016/j.jalgebra.2016.07.001⟩. ⟨hal-00839697⟩