Tomás Aguilar-Fraga ; Jennifer Elder ; Rebecca E. Garcia ; Kimberly P. Hadaway ; Pamela E. Harris et al. - Interval and -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 -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 mn parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter 0, we then consider the subset of interval rational parking functions in which each car parks at most spots away from their initial preference. We call these -interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers mn and . We also establish formulas for the number of nondecreasing -interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between -interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing =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 mn bars. This readily implies enumerative formulas. Further, in the case where =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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