Rigged Horse Numbers and their Modular Periodicity
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917854072274944 |
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| author | Schreyer, Benjamin |
| author_facet | Schreyer, Benjamin |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03799 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigged Horse Numbers and their Modular Periodicity Schreyer, Benjamin Combinatorics 06A05 (Primary) G.2.1 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. |
| title | Rigged Horse Numbers and their Modular Periodicity |
| topic | Combinatorics 06A05 (Primary) G.2.1 |
| url | https://arxiv.org/abs/2409.03799 |