The free and parking quasi-symmetrizing actions

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 ${\text{FQSym}}^{\ast }$ and ${\text{PQSym}}^{\ast }$. 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 ${\text{PQSym}}^{\ast }$. 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.

18 pages