Counting $\ell$-interval Fubini rankings through their parking outcome

Fubini rankings with $n$ competitors are $n$-tuples with entries in $[n]=\{1,2,3,\ldots, n\}$ that encode the conclusion of a race that allows ties. Since Fubini rankings are parking functions, we can study their parking outcomes, which are permutations encoding the final parking order of the cars using the Fubini ranking as a preference list. We establish that the number of Fubini rankings with $n$ competitors having a fixed parking outcome $π$ is given by $2^{n-k}$, where $k$ denotes the number of runs in $π$. We then use this formula to give a new proof for the number of Fubini rankings, which is given by the Fubini numbers. We also consider the set of $\ell$-interval Fubini rankings with $n$ competitors, which are Fubini rankings where at most $\ell+1$ competitors tie at any rank. We show that the number of $\ell$-interval Fubini rankings with $n$ competitors having a fixed parking outcome $π$ is given by a product of a power of two and a product of $\ell$-Pingala numbers, where these factors depend only on the lengths of the runs that make up the parking outcome $π$. The $1$-interval Fubini rankings are known as unit Fubini rankings, and we show that the number of unit Fubini rankings having a fixed parking outcome $π$ is given by a product of Fibonacci numbers indexed by the lengths of the runs in $π$. We use these results to give a formula for the number of $\ell$-interval Fubini rankings with $n$ competitors for all $\ell\in[n]$. We conclude with some directions for further study.