Hopf algebras of parking functions and decorated planar trees

TL;DR

This paper introduces three new combinatorial Hopf algebras based on planar trees and parking functions, revealing their algebraic properties and connections to existing structures like PQSym and Tamari orders.

Contribution

The authors construct novel Hopf algebras on planar trees and parking functions, analyze their algebraic properties, and establish links to known combinatorial Hopf algebras.

Findings

The new algebras are bidendriform, free, cofree, and self-dual.

They define partial orders related to Tamari and m-Tamari orders.

Explicit formulas for multiplication and antipode are provided.

Abstract

We construct three new combinatorial Hopf algebras based on the Loday-Ronco operations on planar binary trees. The first and second algebras are defined on planar trees and labeled planar trees extending the Loday-Ronco and Malvenuto-Reutenauer Hopf algebras respectively. We show that the latter is bidendriform which implies that is also free, cofree, and self-dual. The third algebra involves a new visualization of parking functions as decorated binary trees; it is also bidendriform, free, cofree, and self-dual, and therefore abstractly isomorphic to the algebra PQSym of Novelli and Thibon. We define partial orders on the objects indexing each of these three Hopf algebras, one of which, when restricting to (m+1)-ary trees, coarsens the m-Tamari order of Bergeron and Pr\'eville-Ratelle. We show that multiplication of dual fundamental basis elements are given by intervals in each of these orders. Finally, we use an axiomatized version of the techniques of Aguiar and Sottile on the Malvenuto-Reutenauer Hopf algebra to define a monomial basis on each of our Hopf algebras, and to show that comultiplication is cofree on the monomial elements. This in particular, implies the cofreeness of the Hopf algebra on planar trees. We also find explicit positive formulas for the multiplication on monomial basis and a cancellation-free and grouping-free formula for the antipode of monomial elements.