Interval and $\ell$-interval Rational Parking Functions

TL;DR

This paper introduces and enumerates interval and $ ext{\ell}$-interval rational parking functions, generalizing parking functions with interval preferences and establishing connections to Fibonacci numbers and barred arrangements.

Contribution

It generalizes parking functions to interval rational parking functions, provides enumeration formulas, recursive relations, and explores their connections to Fubini rankings and Fibonacci numbers.

Findings

Derived formulas for counting interval rational parking functions.

Established recursive formulas for $ ext{\ell}$-interval rational parking functions.

Connected $ ext{\ell}$-interval parking functions to Fubini rankings and Fibonacci numbers.

Abstract

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.