Geodesic
Interval and $\ell$-interval Rational Parking Functions
Discrete mathematics & theoretical computer science, Permutation Patterns 2023, Tome 26 (2024) no. 1.

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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.
DOI : 10.46298/dmtcs.12598
Classification : 05A05, 05A15, 05A18, 05A19 
@article{DMTCS_2024_26_1_a6,
     author = {Aguilar-Fraga, Tom\'as and Elder, Jennifer and Garcia, Rebecca E. and Hadaway, Kimberly P. and Harris, Pamela E. and Harry, Kimberly J. and Hogan, Imhotep B. and Johnson, Jakeyl and Kretschmann, Jan and Lawson-Chavanu, Kobe and Mori, J. Carlos Mart{\'\i}nez and Monroe, Casandra D. and Qui\~nonez, Daniel and Tolson III, Dirk and Williams II, Dwight Anderson},
     title = {Interval and $\ell$-interval {Rational} {Parking} {Functions}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2024},
     doi = {10.46298/dmtcs.12598},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12598/}
}
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AU  - Hogan, Imhotep B.
AU  - Johnson, Jakeyl
AU  - Kretschmann, Jan
AU  - Lawson-Chavanu, Kobe
AU  - Mori, J. Carlos Martínez
AU  - Monroe, Casandra D.
AU  - Quiñonez, Daniel
AU  - Tolson III, Dirk
AU  - Williams II, Dwight Anderson
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JO  - Discrete mathematics & theoretical computer science
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%A Elder, Jennifer
%A Garcia, Rebecca E.
%A Hadaway, Kimberly P.
%A Harris, Pamela E.
%A Harry, Kimberly J.
%A Hogan, Imhotep B.
%A Johnson, Jakeyl
%A Kretschmann, Jan
%A Lawson-Chavanu, Kobe
%A Mori, J. Carlos Martínez
%A Monroe, Casandra D.
%A Quiñonez, Daniel
%A Tolson III, Dirk
%A Williams II, Dwight Anderson
%T Interval and $\ell$-interval Rational Parking Functions
%J Discrete mathematics & theoretical computer science
%D 2024
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12598/
%R 10.46298/dmtcs.12598
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%F DMTCS_2024_26_1_a6
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. Interval and $\ell$-interval Rational Parking Functions. Discrete mathematics & theoretical computer science, Permutation Patterns 2023, Tome 26 (2024) no. 1. doi : 10.46298/dmtcs.12598. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.12598/

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