Tomás Aguilar-Fraga ; Jennifer Elder ; Rebecca E. Garcia ; Kimberly P. Hadaway ; Pamela E. Harris et al. - Interval and $\ell$-interval Rational Parking Functions

dmtcs:12598 - Discrete Mathematics & Theoretical Computer Science, November 4, 2024, vol. 26:1, Permutation Patterns 2023 - https://doi.org/10.46298/dmtcs.12598
Interval and $\ell$-interval Rational Parking FunctionsArticle

Authors: 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 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.


    Volume: vol. 26:1, Permutation Patterns 2023
    Section: Combinatorics
    Published on: November 4, 2024
    Accepted on: August 30, 2024
    Submitted on: November 27, 2023
    Keywords: Mathematics - Combinatorics,05A05, 05A15, 05A18, 05A19

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