The free and parking quasi-symmetrizing actions
Adrien Segovia
TL;DR
This work constructs two invariant Hopf algebras, $FQSym^*$ and $PQSym^*$, via free and parking quasi-symmetrizing actions of the infinite symmetric group on words, and provides polynomial realizations, orbit-sum bases, and fundamental (co)product structures. It introduces a parameterized interpolation, generating a nested chain of Hopf subalgebras ${\textbf{PQSym}^*}^r$ that connect $PQSym^*$ to its limits, with explicit Hilbert series and bases indexed by $r$-bi-words. In the special case $r=\infty$, the dimensions of homogeneous components enumerate rooted labeled non-planar trees with maximal decreasing subtrees forming a chain, linking parking-function combinatorics to classical tree enumeration via bijections and known formulas. Overall, the paper furnishes invariant-based, combinatorially rich realizations of quasi-symmetric Hopf algebras and unveils deep connections between parking structures, $r$-interpolations, and tree enumerations.
Abstract
We define two actions of the infinite symmetric group on the set of words on positive integers, called the free and parking quasi-symmetrizing actions, whose invariants are respectively the elements of the Hopf algebras $\textbf{FQSym}^*$ and $\textbf{PQSym}^*$. We study in depth the parking quasi-symmetrizing action by generalizing it to actions with a parameter $r\in(\mathbb{N}\setminus \{0\} )\bigcup\{\infty\}$. We prove that the spaces of the invariants under these $r$-actions form an infinite chain of nested graded Hopf subalgebras of $\textbf{PQSym}^*$. We give some properties of these Hopf algebras including their Hilbert series, a basis, and formulas for their product and coproduct. Finally we look more closely at the case $r=\infty$, obtaining enumerative results related to trees with maximal decreasing subtrees of given sizes.