Balanced Steinhaus triangles
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913978877214720 |
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| author | Chappelon, Jonathan |
| author_facet | Chappelon, Jonathan |
| contents | A Steinhaus triangle modulo $m$ is a finite down-pointing triangle of elements in the finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ satisfying the same local rule as the standard Pascal triangle modulo $m$. A Steinhaus triangle modulo $m$ is said to be balanced if it contains all the elements of $\mathbb{Z}/m\mathbb{Z}$ with the same multiplicity. In this paper, the existence of infinitely many balanced Steinhaus triangles modulo $m$, for any positive integer $m$, is shown. This is achieved by considering periodic triangles generated from interlaced arithmetic progressions. This positively answers a weak version of a problem, due to John C. Molluzzo in 1978, that has remained unsolved to date for the even values of $m\geqslant 12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Balanced Steinhaus triangles Chappelon, Jonathan Combinatorics Discrete Mathematics Number Theory 05B30, 11B75, 11B25, 11B65, 11B50 A Steinhaus triangle modulo $m$ is a finite down-pointing triangle of elements in the finite cyclic group $\mathbb{Z}/m\mathbb{Z}$ satisfying the same local rule as the standard Pascal triangle modulo $m$. A Steinhaus triangle modulo $m$ is said to be balanced if it contains all the elements of $\mathbb{Z}/m\mathbb{Z}$ with the same multiplicity. In this paper, the existence of infinitely many balanced Steinhaus triangles modulo $m$, for any positive integer $m$, is shown. This is achieved by considering periodic triangles generated from interlaced arithmetic progressions. This positively answers a weak version of a problem, due to John C. Molluzzo in 1978, that has remained unsolved to date for the even values of $m\geqslant 12$. |
| title | Balanced Steinhaus triangles |
| topic | Combinatorics Discrete Mathematics Number Theory 05B30, 11B75, 11B25, 11B65, 11B50 |
| url | https://arxiv.org/abs/2508.05159 |