Enumerating Vector Parking Functions and their Outcomes Based on Specified Lucky Cars
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915488257277952 |
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| author | Ferreri, Melanie Harris, Pamela E. Martinez, Lucy Swartz, Eric |
| author_facet | Ferreri, Melanie Harris, Pamela E. Martinez, Lucy Swartz, Eric |
| contents | In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or $\boldsymbol{u}$-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on $\boldsymbol{u}$, we enumerate outcomes of $\boldsymbol{u}$-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating $\boldsymbol{u}$-parking functions themselves according to their set of lucky cars. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumerating Vector Parking Functions and their Outcomes Based on Specified Lucky Cars Ferreri, Melanie Harris, Pamela E. Martinez, Lucy Swartz, Eric Combinatorics 05A05, 05A10, 05A15 In a parking function, a car is considered lucky if it is able to park in its preferred spot. Extending work of Harris and Martinez, we enumerate outcomes of parking functions with a fixed set of lucky cars. We then consider a generalization of parking functions known as vector parking functions or $\boldsymbol{u}$-parking functions, in which a nonnegative integer capacity is given to each parking spot in the street. With certain restrictions on $\boldsymbol{u}$, we enumerate outcomes of $\boldsymbol{u}$-parking functions with a fixed set of lucky cars or with a fixed number of lucky cars. We also count outcomes according to which spots contain lucky cars, and give formulas for enumerating $\boldsymbol{u}$-parking functions themselves according to their set of lucky cars. |
| title | Enumerating Vector Parking Functions and their Outcomes Based on Specified Lucky Cars |
| topic | Combinatorics 05A05, 05A10, 05A15 |
| url | https://arxiv.org/abs/2508.13917 |