The Hopf algebra of integer binary relations

Vincent Pilaud (1) , Viviane Pons (2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 GALaC - LRI - Graphes, Algorithmes et Combinatoire (LRI)

We construct a Hopf algebra on integer binary relations that contains under the same roof several well-known Hopf algebras related to the permutahedra and the associahedra: the Malvenuto-Reutenauer algebra on permutations, the Loday-Ronco algebra on planar binary trees, and the Chapoton algebras on ordered partitions and on Schröder trees. We also derive from our construction new Hopf structures on intervals of the weak order on permutations and of the Tamari order on binary trees.

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Vincent Pilaud, Viviane Pons. The Hopf algebra of integer binary relations. Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 1, European Mathematical Society Publishing House, pp.299-344, 2020, ⟨10.4171/204-1/8⟩. ⟨hal-02469753⟩