Interval and $\ell$-interval Rational Parking Functions
Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for...
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Discrete Mathematics & Theoretical Computer Science
2024-11-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | http://dmtcs.episciences.org/12598/pdf |
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| author | Tomás Aguilar-Fraga Jennifer Elder Rebecca E. Garcia Kimberly P. Hadaway Pamela E. Harris Kimberly J. Harry Imhotep B. Hogan Jakeyl Johnson Jan Kretschmann Kobe Lawson-Chavanu J. Carlos Martínez Mori Casandra D. Monroe Daniel Quiñonez Dirk Tolson III Dwight Anderson Williams II |
| author_facet | Tomás Aguilar-Fraga Jennifer Elder Rebecca E. Garcia Kimberly P. Hadaway Pamela E. Harris Kimberly J. Harry Imhotep B. Hogan Jakeyl Johnson Jan Kretschmann Kobe Lawson-Chavanu J. Carlos Martínez Mori Casandra D. Monroe Daniel Quiñonez Dirk Tolson III Dwight Anderson Williams II |
| author_sort | Tomás Aguilar-Fraga |
| collection | DOAJ |
| description | Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers. |
| format | Article |
| id | doaj-art-b109e3807b8f468c8bf1c412af2cf6e3 |
| institution | DOAJ |
| issn | 1365-8050 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj-art-b109e3807b8f468c8bf1c412af2cf6e32025-08-20T02:51:42ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502024-11-01vol. 26:1, Permutation...Combinatorics10.46298/dmtcs.1259812598Interval and $\ell$-interval Rational Parking FunctionsTomás Aguilar-FragaJennifer ElderRebecca E. GarciaKimberly P. HadawayPamela E. HarrisKimberly J. HarryImhotep B. HoganJakeyl JohnsonJan KretschmannKobe Lawson-ChavanuJ. Carlos Martínez MoriCasandra D. MonroeDaniel QuiñonezDirk Tolson IIIDwight Anderson Williams IIInterval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with $n$ cars and $m\geq n$ parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter $\ell\geq 0$, we then consider the subset of interval rational parking functions in which each car parks at most $\ell$ spots away from their initial preference. We call these $\ell$-interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers $m\geq n$ and $\ell$. We also establish formulas for the number of nondecreasing $\ell$-interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between $\ell$-interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing $\ell=1$, and establish that the set of $1$-interval rational parking functions with $n$ cars and $m$ spots are in bijection with the set of barred preferential arrangements of $[n]$ with $m-n$ bars. This readily implies enumerative formulas. Further, in the case where $\ell=1$, we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.http://dmtcs.episciences.org/12598/pdfmathematics - combinatorics05a05, 05a15, 05a18, 05a19 |
| spellingShingle | Tomás Aguilar-Fraga Jennifer Elder Rebecca E. Garcia Kimberly P. Hadaway Pamela E. Harris Kimberly J. Harry Imhotep B. Hogan Jakeyl Johnson Jan Kretschmann Kobe Lawson-Chavanu J. Carlos Martínez Mori Casandra D. Monroe Daniel Quiñonez Dirk Tolson III Dwight Anderson Williams II Interval and $\ell$-interval Rational Parking Functions Discrete Mathematics & Theoretical Computer Science mathematics - combinatorics 05a05, 05a15, 05a18, 05a19 |
| title | Interval and $\ell$-interval Rational Parking Functions |
| title_full | Interval and $\ell$-interval Rational Parking Functions |
| title_fullStr | Interval and $\ell$-interval Rational Parking Functions |
| title_full_unstemmed | Interval and $\ell$-interval Rational Parking Functions |
| title_short | Interval and $\ell$-interval Rational Parking Functions |
| title_sort | interval and ell interval rational parking functions |
| topic | mathematics - combinatorics 05a05, 05a15, 05a18, 05a19 |
| url | http://dmtcs.episciences.org/12598/pdf |
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