Lucky Cars in Fubini Rankings and Unit Fubini Rankings

TL;DR

This paper explores the combinatorial properties and enumeration of lucky cars in Fubini and unit Fubini rankings, linking these concepts to ordered set partitions and compositions using advanced enumeration techniques.

Contribution

It provides new combinatorial characterizations, enumeration formulas, and connections for lucky cars in Fubini rankings, employing diverse enumeration methods.

Findings

Derived enumeration formulas for lucky cars in Fubini rankings.

Established connections between rankings and ordered set partitions.

Applied advanced combinatorial techniques like generating functions and Zeilberger's method.

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.