Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.03799 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.