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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2502.07926 |
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| _version_ | 1866912229777997824 |
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| author | Segovia, Adrien |
| author_facet | Segovia, Adrien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The free and parking quasi-symmetrizing actions Segovia, Adrien Combinatorics Rings and Algebras 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. |
| title | The free and parking quasi-symmetrizing actions |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2502.07926 |