Rigged Horse Numbers and their Modular Periodicity

TL;DR

This paper explores the properties of r-Fubini numbers, expressing them through shifted sequences and signed Stirling numbers, and demonstrates their eventual modular periodicity with periods linked to the Carmichael function.

Contribution

It introduces a novel shift operator counting method and establishes the modular periodicity of r-Fubini numbers, connecting their maximum period to the Carmichael function.

Findings

r-Fubini numbers can be expressed as sums of shifted Fubini sequences.

The modular periodicity of r-Fubini numbers is demonstrated.

Maximum period equals the Carmichael function of the modulus.

Abstract

The Fubini numbers count the permutations of horse racing where ties are possible. The closely related $r$-horse numbers count the finishes of a horse race where some subset of $r$ horses agree to finish the race in a specific relative strong ordering. We express the $r$-Fubini numbers as a sum of $r$ index-shifted sequences of Fubini numbers weighted with the signed Stirling numbers of the first kind. We use a novel shift operator counting. Further, we demonstrate the eventual modular periodicity of $r$-Fubini numbers. Their maximum period is determined to be the Carmichael function of the modulus. The maximum period occurs in the case of an odd modulus for Fubini numbers.