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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.03651 |
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| _version_ | 1866916955154284544 |
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| author | Snider, Lauren Yan, Catherine |
| author_facet | Snider, Lauren Yan, Catherine |
| contents | Graphical parking functions, or $G$-parking functions, are a generalization of classical parking functions which depend on a connected multigraph $G$ having a distinguished root vertex. Gaydarov and Hopkins characterized the relationship between $G$-parking functions and another vector-dependent generalization of parking functions, the $\boldsymbol{u}$-parking functions. The crucial component of their result was their classification of all graphs $G$ whose $G$-parking functions are invariant under action by the symmetric group $\mathfrak{S}_n$, where $n+1$ is the order of $G$. In this work, we present a 2-dimensional analogue of Gaydarov and Hopkins' results by characterizing the overlap between $G$-parking functions and 2-dimensional $\boldsymbol{U}$-parking functions, i.e., pairs of integer sequences whose order statistics are bounded by certain weights along lattice paths in the plane. Our key result is a total classification of all $G$ whose set of $G$-parking functions is $(\mathfrak{S}_p \times \mathfrak{S}_q)$-invariant, where $p+q+1$ is the order of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_03651 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | ($\mathfrak{S}_p \times \mathfrak{S}_q$)-Invariant Graphical Parking Functions Snider, Lauren Yan, Catherine Combinatorics 05C57, 05C30, 05E18 Graphical parking functions, or $G$-parking functions, are a generalization of classical parking functions which depend on a connected multigraph $G$ having a distinguished root vertex. Gaydarov and Hopkins characterized the relationship between $G$-parking functions and another vector-dependent generalization of parking functions, the $\boldsymbol{u}$-parking functions. The crucial component of their result was their classification of all graphs $G$ whose $G$-parking functions are invariant under action by the symmetric group $\mathfrak{S}_n$, where $n+1$ is the order of $G$. In this work, we present a 2-dimensional analogue of Gaydarov and Hopkins' results by characterizing the overlap between $G$-parking functions and 2-dimensional $\boldsymbol{U}$-parking functions, i.e., pairs of integer sequences whose order statistics are bounded by certain weights along lattice paths in the plane. Our key result is a total classification of all $G$ whose set of $G$-parking functions is $(\mathfrak{S}_p \times \mathfrak{S}_q)$-invariant, where $p+q+1$ is the order of $G$. |
| title | ($\mathfrak{S}_p \times \mathfrak{S}_q$)-Invariant Graphical Parking Functions |
| topic | Combinatorics 05C57, 05C30, 05E18 |
| url | https://arxiv.org/abs/2305.03651 |