On vector parking functions and q-analogue
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913344842104832 |
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| author | Yang, Wenkai |
| author_facet | Yang, Wenkai |
| contents | In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H. Yan systematically investigated the properties of rational parking functions. Subsequently, a series of Context-free grammars possessing the requisite property were introduced by William Y.C. Chen and Harold R.L. Yang in 2021. %An Abelian-type identity is derived from a comparable methodology and grammatical framework. %Leveraging a comparable methodology and grammatical framework, an Abelian-type identity is derived herein. In this paper, I discuss generalized parking functions in terms of grammars. The primary result is to obtain the q-analogue about the number of '1's in certain vector parking functions with the assistance of grammars. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04954 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On vector parking functions and q-analogue Yang, Wenkai Combinatorics 05A15, 05A19, 05A30 In 2000, it was demonstrated that the set of $x$-parking functions of length $n$, where $x$=($a,b,...,b$) $\in \mathbbm{N}^n$, is equivalent to the set of rooted multicolored forests on [$n$]=\{1,...,$n$\}. In 2020, Yue Cai and Catherine H. Yan systematically investigated the properties of rational parking functions. Subsequently, a series of Context-free grammars possessing the requisite property were introduced by William Y.C. Chen and Harold R.L. Yang in 2021. %An Abelian-type identity is derived from a comparable methodology and grammatical framework. %Leveraging a comparable methodology and grammatical framework, an Abelian-type identity is derived herein. In this paper, I discuss generalized parking functions in terms of grammars. The primary result is to obtain the q-analogue about the number of '1's in certain vector parking functions with the assistance of grammars. |
| title | On vector parking functions and q-analogue |
| topic | Combinatorics 05A15, 05A19, 05A30 |
| url | https://arxiv.org/abs/2405.04954 |