Hopf algebras of parking functions and decorated planar trees
Nantel Bergeron, Rafael S. Gonz\'alez D'Le\'on, Shu Xiao Li and, C.Y. Amy Pang, Yannic Vargas

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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Graph Theory Research · Commutative Algebra and Its Applications
