Interval and $\ell$-interval Rational Parking Functions
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| Main Authors: | , , , , , , , , , , , , , , |
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| Format: | Preprint |
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2023
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| author | Aguilar-Fraga, Tomás Elder, Jennifer Garcia, Rebecca E. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Hogan, Imhotep B. Johnson, Jakeyl Kretschmann, Jan Lawson-Chavanu, Kobe Mori, J. Carlos Martínez Monroe, Casandra D. Quiñonez, Daniel Tolson III, Dirk Williams II, Dwight Anderson |
| author_facet | Aguilar-Fraga, Tomás Elder, Jennifer Garcia, Rebecca E. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Hogan, Imhotep B. Johnson, Jakeyl Kretschmann, Jan Lawson-Chavanu, Kobe Mori, J. Carlos Martínez Monroe, Casandra D. Quiñonez, Daniel Tolson III, Dirk Williams II, Dwight Anderson |
| contents | 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 | Preprint |
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arxiv_https___arxiv_org_abs_2311_14055 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Interval and $\ell$-interval Rational Parking Functions Aguilar-Fraga, Tomás Elder, Jennifer Garcia, Rebecca E. Hadaway, Kimberly P. Harris, Pamela E. Harry, Kimberly J. Hogan, Imhotep B. Johnson, Jakeyl Kretschmann, Jan Lawson-Chavanu, Kobe Mori, J. Carlos Martínez Monroe, Casandra D. Quiñonez, Daniel Tolson III, Dirk Williams II, Dwight Anderson Combinatorics 05A05, 05A15, 05A18, 05A19 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. |
| title | Interval and $\ell$-interval Rational Parking Functions |
| topic | Combinatorics 05A05, 05A15, 05A18, 05A19 |
| url | https://arxiv.org/abs/2311.14055 |