Vacillating parking functions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912001521876992 |
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| author | Fang, Bruce Harris, Pamela E. Kamau, Brian M. Wang, David |
| author_facet | Fang, Bruce Harris, Pamela E. Kamau, Brian M. Wang, David |
| contents | For any integers $1\leq k\leq n$, we introduce a new family of parking functions called $k$-vacillating parking functions of length $n$. The parking rule for $k$-vacillating parking functions allows a car with preference $p$ to park in the first available spot in encounters among the parking spots numbered $p$, $p-k$, and $p+k$ (in that order and if those spots exists). In this way, $k$-vacillating parking functions are a modification of Naples parking functions, which allow for backwards movement of a car, and of $\ell$-interval parking functions, which allow a car to park in its preference or up to $\ell$ spots in front of its preference. Among our results, we establish a combinatorial interpretation for the numerator of the $n$th convergent of the continued fraction of $\sqrt{2}$, as the number of non-decreasing $1$-vacillating parking functions of length~$n$. Our main result gives a product formula for the enumeration of $k$-vacillating parking functions of length $n$ based on the number of $1$-vacillating parking functions of smaller length. We conclude with some directions for further research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vacillating parking functions Fang, Bruce Harris, Pamela E. Kamau, Brian M. Wang, David Combinatorics 05A05, 05A15 For any integers $1\leq k\leq n$, we introduce a new family of parking functions called $k$-vacillating parking functions of length $n$. The parking rule for $k$-vacillating parking functions allows a car with preference $p$ to park in the first available spot in encounters among the parking spots numbered $p$, $p-k$, and $p+k$ (in that order and if those spots exists). In this way, $k$-vacillating parking functions are a modification of Naples parking functions, which allow for backwards movement of a car, and of $\ell$-interval parking functions, which allow a car to park in its preference or up to $\ell$ spots in front of its preference. Among our results, we establish a combinatorial interpretation for the numerator of the $n$th convergent of the continued fraction of $\sqrt{2}$, as the number of non-decreasing $1$-vacillating parking functions of length~$n$. Our main result gives a product formula for the enumeration of $k$-vacillating parking functions of length $n$ based on the number of $1$-vacillating parking functions of smaller length. We conclude with some directions for further research. |
| title | Vacillating parking functions |
| topic | Combinatorics 05A05, 05A15 |
| url | https://arxiv.org/abs/2402.02538 |