Lucky Cars in Fubini Rankings and Unit Fubini Rankings
Camilo Barreto, Melissa Beerbower, Jennifer Elder, Pamela E. Harris, Lucy Martinez, José L. Ramírez, Samuel Ramírez, Grant Shirley, Julio C. Vásquez
TL;DR
This work analyzes lucky cars in two related combinatorial families—Fubini rankings $FR_n$ and unit Fubini rankings $UFR_n$—and their unit-interval parking function counterpart $UPF_n$, establishing precise enumerative and structural results. It shows that in $FR_n$ the lucky cars correspond to the first in each tie block, yielding $f_{\mathrm{FR}}(n,k)=k!\,S(n,k)$ and the lucky polynomial $L_{\mathrm{FR}_n}(q)=\sum_{k} k!\,S(n,k)\,q^{k}$, with a compact EGF $\frac{1}{1-(e^{x}-1)q}$, while the weakly increasing case collapses to binomial counts and a simple polynomial $L_{\mathrm{FR}_n^{\uparrow}}(q)=q(q+1)^{n-1}$. The unit-Fubini results follow via a Hadaway bijection with $UPF_n$, giving closed forms and recurrences for $f_{\mathrm{UFR}}(n,k)$ and $f_{\mathrm{UFR}}^{\uparrow}(n,k)$, as well as explicit expressions for the corresponding lucky polynomials and generating functions; fixed-set enumerations are obtained for both unit and non-unit variants. The paper also develops detailed weakly increasing cases, highlighting Fibonacci-type and binomial structures, and characterizes feasible lucky sets under weakly increasing constraints. Together, these results deepen the connections between parking-function-like objects, restricted partitions, and algebraic/combinatorial techniques such as exponential generating functions and Zeilberger’s telescoping method, with asymptotic insights for the mean number of lucky cars. The work provides a cohesive framework for enumerating and analyzing lucky statistics across several intertwined combinatorial families, with multiple avenues for generalization and further study.
Abstract
We study lucky cars in subsets of parking functions, called Fubini rankings and unit Fubini rankings. A Fubini ranking is a sequence of nonnegative integers that encodes a valid ranking of competitors, where ties are allowed. A car (or competitor) is said to be lucky if it is the first instance of that rank appearing in the sequence. We present combinatorial characterizations and enumeration formulas for lucky cars in both Fubini rankings and unit Fubini rankings, and establish connections between these objects and ordered set partitions, as well as integer compositions. To obtain our results, we use several techniques to enumerate statistics over these families of objects. In particular, we employ generating functions, bijective and combinatorial arguments, recurrence relations, and Zeilberger's creative telescoping method.