Restricted Fubini Rankings and Restricted Unit Interval Parking Functions
Camilo Barreto, Pamela Harris, José L. Ramírez, Samuel Ramírez, Julio C. Vasquez
TL;DR
This paper develops three notions of S-restriction for Fubini rankings and unit interval parking functions, motivated by their correspondence with ordered set partitions. It establishes a central type-1 bijection between S-restricted FR and UPF, and derives exponential generating functions that capture the enumeration across arbitrary S, with specialized results for even/odd block structures, cyclical adjacencies, and exceedances. It extends the framework to type-2 and type-3 restrictions, providing general bijections to UPF variants, explicit singleton and finite-S counting formulas, and alternative generating-function perspectives. Collectively, the work illuminates how restricted combinatorial species related to parking functions and rankings encode ordered partitions and permits precise enumerations via multinomial and EGF techniques. The results deepen connections between parking functions, Fubini rankings, and ordered partitions, suggesting rich avenues for q-analogues and algebraic interpretations.
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.