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

Bjorn Andreas Ager-Hart, Melissa Beerbower, Pamela E. Harris, Joakim Jakovleski, Matt McClinton

Abstract

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.

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

Abstract

Fubini rankings with competitors are -tuples with entries in 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 competitors having a fixed parking outcome is given by , where denotes the number of runs in . 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 -interval Fubini rankings with competitors, which are Fubini rankings where at most competitors tie at any rank. We show that the number of -interval Fubini rankings with competitors having a fixed parking outcome is given by a product of a power of two and a product of -Pingala numbers, where these factors depend only on the lengths of the runs that make up the parking outcome . The -interval Fubini rankings are known as unit Fubini rankings, and we show that the number of unit Fubini rankings having a fixed parking outcome is given by a product of Fibonacci numbers indexed by the lengths of the runs in . We use these results to give a formula for the number of -interval Fubini rankings with competitors for all . We conclude with some directions for further study.
Paper Structure (9 sections, 24 theorems, 76 equations, 2 figures)

This paper contains 9 sections, 24 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Enumerating Fubini Rankings by their Parking Outcome
  3. Enumerating Unit Fubini Rankings by their Parking Outcome
  4. Enumerating $\ell$-interval Fubini Rankings by their Outcome
  5. Future Work
  6. $(m,n)$-Fubini Rankings
  7. $b$-Fubini Rankings
  8. Fubini Rankings with a Fixed Maximum Rank
  9. $k$-Restricted Fubini Rankings

Key Result

Lemma 2.1

Let $\alpha=(a_1,a_2,\ldots, a_n)\in[n]^n$ and denote the weakly increasing rearrangement of $\alpha$ by $\alpha^\uparrow=(a'_1,a'_2,\ldots, a'_n)$, where $a_i'\leq a_{i+1}'$ for all $i\in[n-1]$. Then $\alpha\in \mathrm{FR}_{n}$ if and only if for any $i\in [n]$, either $a'_i=i$ or $a'_i=a'_{i-1}$.

Figures (2)

  • Figure 1: The 13 possible results of three competitors competing in a race allowing ties.
  • Figure 2: Parking cars using the preferences in $\alpha=(4,1,1,1,5)$. Car image designed by Freepik and callout designed by macrovector / Freepik.

Theorems & Definitions (60)

  • Remark 2.1
  • Definition 2.1
  • Lemma 2.1: Characterization of Fubini rankings
  • proof
  • Example 2.1
  • Lemma 2.2: Theorem 3.13 barreto2025restrictedfubinirankingsrestricted
  • Example 2.2
  • Lemma 2.3: Ranks Lemma
  • proof
  • Proposition 2.1
  • ...and 50 more