Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model
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| Format: | Preprint |
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2025
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| _version_ | 1866915364166696960 |
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| author | Selig, Thomas Zhu, Haoyue |
| author_facet | Selig, Thomas Zhu, Haoyue |
| contents | This work investigates the duality between two discrete dynamical processes: parking functions, and the Abelian sandpile model (ASM). Specifically, we are interested in the extension of classical parking functions, called $G$-parking functions, introduced by Postnikov and Shapiro in 2004. $G$-parking functions are in bijection with recurrent configurations of the ASM on $G$. In this work, we define a notion of prime $G$-parking functions. These are parking functions that are in a sense "indecomposable". Our notion extends the concept of primeness for classical parking functions, as well as the notion of prime $(p,q)$-parking functions introduced by Armon et al. in recent work. We show that from the ASM perspective, prime $G$-parking functions correspond to certain configurations of the ASM, which we call strongly recurrent. We study this new connection on a number of graph families, including wheel graphs, complete graphs, complete multi-partite graphs, and complete split graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model Selig, Thomas Zhu, Haoyue Combinatorics 05A19 (Primary), 05A15, 05C30 (Secondary) This work investigates the duality between two discrete dynamical processes: parking functions, and the Abelian sandpile model (ASM). Specifically, we are interested in the extension of classical parking functions, called $G$-parking functions, introduced by Postnikov and Shapiro in 2004. $G$-parking functions are in bijection with recurrent configurations of the ASM on $G$. In this work, we define a notion of prime $G$-parking functions. These are parking functions that are in a sense "indecomposable". Our notion extends the concept of primeness for classical parking functions, as well as the notion of prime $(p,q)$-parking functions introduced by Armon et al. in recent work. We show that from the ASM perspective, prime $G$-parking functions correspond to certain configurations of the ASM, which we call strongly recurrent. We study this new connection on a number of graph families, including wheel graphs, complete graphs, complete multi-partite graphs, and complete split graphs. |
| title | Prime graphical parking functions and strongly recurrent configurations of the Abelian sandpile model |
| topic | Combinatorics 05A19 (Primary), 05A15, 05C30 (Secondary) |
| url | https://arxiv.org/abs/2506.23237 |