Unit interval parking functions and the $r$-Fubini numbers

Abstract: We recall that unit interval parking functions of length $n$ are a subset of parking functions in which every car parks in its preference or in the spot after its preference, and Fubini rankings of length $n$ are rankings of $n$ competitors allowing for ties. We present an independent proof of a result of Hadaway, which establishes that unit interval parking functions and Fubini rankings are in bijection. We also prove that the cardinality of these sets are given by Fubini numbers. In addition, we give a complete characterization of unit interval parking functions by determining when a rearrangement of a unit interval parking function is again a unit interval parking function. This yields an identity for the Fubini numbers as a sum of multinomials over compositions. Moreover, we introduce a generalization of Fubini rankings, which we call the $r$-Fubini rankings of length $n+r$. We show that this set is in bijection with unit interval parking functions of length $n+r$ where the first $r$ cars have distinct preferences. We conclude by establishing that these sets are enumerated by the $r$-Fubini numbers.