Restricted Fubini Rankings and Restricted Unit Interval Parking Functions

TL;DR

This paper explores restricted classes of Fubini rankings and unit interval parking functions, establishing enumeration, bijections, and generating functions, with connections to permutations and set partitions.

Contribution

It introduces new restricted families of these combinatorial objects, provides enumeration formulas, bijections, and links to permutation statistics and set partition properties.

Findings

Enumerative formulas for restricted families

Bijections between restricted objects

Connections with permutation exceedances and set partition adjacencies

Abstract

We study three natural types of restrictions on Fubini rankings and unit interval parking functions, which are motivated by their correspondence with ordered set partitions. For each restriction type, we define the corresponding subset of Fubini rankings and unit interval parking functions, establish enumerative results, and provide bijections between the restricted families. We also obtain exponential generating functions and combinatorial interpretations, including connections with exceedances in permutations and with the absence of cyclical adjacencies in set partitions.