The free and parking quasi-symmetrizing actions

TL;DR

This paper introduces two actions of the infinite symmetric group on words, explores their invariants forming nested Hopf subalgebras, and provides algebraic and enumerative insights, especially for the case when the parameter is infinity.

Contribution

It defines and analyzes the free and parking quasi-symmetrizing actions, generalizes the parking action with a parameter, and studies the structure of their invariant Hopf algebras.

Findings

Invariants form nested graded Hopf subalgebras of PQSym*.

Provides bases, Hilbert series, and product/coproduct formulas for these subalgebras.

Enumerative results related to trees with maximal decreasing subtrees for r=∞.

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.