Pullback parking functions
We introduce a generalization of parking functions in which cars are limited in their movement backwards and forwards by two nonnegative integer parameters \(k\) and \(\ell\), respectively. In this setting, there are \(n\) spots on a one-way streetand \(m\) cars attempting to park in those spots, an...
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American Journal of Combinatorics
2025-02-01
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| Series: | The American Journal of Combinatorics |
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| Online Access: | https://ajcombinatorics.org/ojs/index.php/AmJC/article/view/21 |
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| author | Jennifer Elder Pamela Harris Lybitina Koene Ilana Lavene Lucy Martinez Molly Oldham |
| author_facet | Jennifer Elder Pamela Harris Lybitina Koene Ilana Lavene Lucy Martinez Molly Oldham |
| author_sort | Jennifer Elder |
| collection | DOAJ |
| description | We introduce a generalization of parking functions in which cars are limited in their movement backwards and forwards by two nonnegative integer parameters \(k\) and \(\ell\), respectively. In this setting, there are \(n\) spots on a one-way streetand \(m\) cars attempting to park in those spots, and \(1\leq m\leq n\). We let \(\alpha=(a_1,a_2,\ldots,a_m)\in[n]^m\) denote the parking preferences for the cars, which enter the street sequentially. Car \(i\) drives to their preference \(a_i\) and parks there if the spot is available. Otherwise, car \(i\) checks up to \(k\) spots behind their preference, parking in the first available spot it encounters if any. If no spots are available, or the car reaches the start of the street, then the car returns to its preference and attempts to park in the first spot it encounters among spots \(a_i+1,a_i+2,\ldots,a_i+\ell\). If car \(i\) fails to park, then parking ceases. If all cars are able to park given the preferences in \(\alpha\), then \(\alpha\) is called a \((k,\ell)\)-pullback \((m,n)\)-parking function. Our main result establishes counts for these parking functions in two ways: counting them based on their final parking outcome (the order in which the cars park on the street), and via a recursive formula. Specializing \(\ell=n-1\), our result gives a new formula for the number of \(k\)-Naples \((m,n)\)-parking functions and further specializing \(m=n\) recovers a formula for the number of \(k\)-Naples parking functions given by Christensen et al. The specialization of \(k=\ell=1\), gives a formula for the number of vacillating \((m,n)\)-parking functions, a generalization of vacillating parking functions studied by Fang et al., and the $m=n$ result answers a problem posed by the authors. We conclude with a few directions for further study. |
| format | Article |
| id | doaj-art-3579cb27f5ed497cbf35210f93ef9657 |
| institution | OA Journals |
| issn | 2768-4202 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | American Journal of Combinatorics |
| record_format | Article |
| series | The American Journal of Combinatorics |
| spelling | doaj-art-3579cb27f5ed497cbf35210f93ef96572025-08-20T02:27:12ZengAmerican Journal of CombinatoricsThe American Journal of Combinatorics2768-42022025-02-01410.63151/amjc.v4i.21Pullback parking functionsJennifer ElderPamela HarrisLybitina KoeneIlana LaveneLucy MartinezMolly OldhamWe introduce a generalization of parking functions in which cars are limited in their movement backwards and forwards by two nonnegative integer parameters \(k\) and \(\ell\), respectively. In this setting, there are \(n\) spots on a one-way streetand \(m\) cars attempting to park in those spots, and \(1\leq m\leq n\). We let \(\alpha=(a_1,a_2,\ldots,a_m)\in[n]^m\) denote the parking preferences for the cars, which enter the street sequentially. Car \(i\) drives to their preference \(a_i\) and parks there if the spot is available. Otherwise, car \(i\) checks up to \(k\) spots behind their preference, parking in the first available spot it encounters if any. If no spots are available, or the car reaches the start of the street, then the car returns to its preference and attempts to park in the first spot it encounters among spots \(a_i+1,a_i+2,\ldots,a_i+\ell\). If car \(i\) fails to park, then parking ceases. If all cars are able to park given the preferences in \(\alpha\), then \(\alpha\) is called a \((k,\ell)\)-pullback \((m,n)\)-parking function. Our main result establishes counts for these parking functions in two ways: counting them based on their final parking outcome (the order in which the cars park on the street), and via a recursive formula. Specializing \(\ell=n-1\), our result gives a new formula for the number of \(k\)-Naples \((m,n)\)-parking functions and further specializing \(m=n\) recovers a formula for the number of \(k\)-Naples parking functions given by Christensen et al. The specialization of \(k=\ell=1\), gives a formula for the number of vacillating \((m,n)\)-parking functions, a generalization of vacillating parking functions studied by Fang et al., and the $m=n$ result answers a problem posed by the authors. We conclude with a few directions for further study.https://ajcombinatorics.org/ojs/index.php/AmJC/article/view/21Parking functioninterval parking functionk-Naples parking functions(m,n)-parking functionpullback parking function |
| spellingShingle | Jennifer Elder Pamela Harris Lybitina Koene Ilana Lavene Lucy Martinez Molly Oldham Pullback parking functions The American Journal of Combinatorics Parking function interval parking function k-Naples parking functions (m,n)-parking function pullback parking function |
| title | Pullback parking functions |
| title_full | Pullback parking functions |
| title_fullStr | Pullback parking functions |
| title_full_unstemmed | Pullback parking functions |
| title_short | Pullback parking functions |
| title_sort | pullback parking functions |
| topic | Parking function interval parking function k-Naples parking functions (m,n)-parking function pullback parking function |
| url | https://ajcombinatorics.org/ojs/index.php/AmJC/article/view/21 |
| work_keys_str_mv | AT jenniferelder pullbackparkingfunctions AT pamelaharris pullbackparkingfunctions AT lybitinakoene pullbackparkingfunctions AT ilanalavene pullbackparkingfunctions AT lucymartinez pullbackparkingfunctions AT mollyoldham pullbackparkingfunctions |