The monomial basis and the $Q$-basis of the Hopf algebra of parking functions

TL;DR

This paper introduces a new Hopf algebra PFSym based on parking functions, constructs monomial and Q-bases, and proves its freeness and isomorphism to PQSym$^{*}$, connecting to known Hopf algebras of trees.

Contribution

It constructs a new Hopf algebra PFSym with monomial and Q-bases, proving its freeness and isomorphism to PQSym$^{*}$, and links to tree-based Hopf algebras.

Findings

PFSym is free and generated by bases analogous to NCSym.

PFSym is isomorphic to PQSym$^{*}$.

Connections to Grossman-Larson Hopf algebras of trees.

Abstract

Consider the vector space $\mathbb{K}\mathcal{P}$ spanned by parking functions. By representing parking functions as labeled digraphs, Hivert, Novelli and Thibon constructed a cocommutative Hopf algebra PQSym$^{*}$ on $\mathbb{K}\mathcal{P}$. The product and coproduct of PQSym$^{*}$ are analogous to the product and coproduct of the Hopf algebra NCSym of symmetric functions in noncommuting variables defined in terms of the power sum basis. In this paper, we view a parking function as a word. We shall construct a Hopf algebra PFSym on $\mathbb{K}\mathcal{P}$ with a formal basis $\{M_a\}$ analogous to the monomial basis of NCSym. By introducing a partial order on parking functions, we transform the basis $\{M_a\}$ to another basis $\{Q_a\}$ via the M\"{o}bius inversion. We prove the freeness of PFSym by finding two free generating sets in terms of the $M$-basis and the $Q$-basis, and we show that PFSym is isomorphic to the Hopf algebra PQSym$^{*}$. It turns out that our construction, when restricted to permutations and non-increasing parking functions, leads to a new way to approach the Grossman-Larson Hopf algebras of ordered trees and heap-ordered trees.