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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2511.19323 |
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| _version_ | 1866914169364676608 |
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| author | Bludov, Mikhail V. Zuev, Nikolai K. |
| author_facet | Bludov, Mikhail V. Zuev, Nikolai K. |
| contents | In this article, we explore the combinatorics of balanced collections. A collection of subsets of the set $[n] = \{1, \dots, n\}$ is called \emph{balanced} if the relative interior of the convex hull of the corresponding characteristic vectors intersects the main diagonal of the $n$-dimensional cube, and it is called \emph{minimal} if it contains no proper balanced subcollections. In particular, we establish both upper and lower bounds for the number of minimal balanced collections. Specifically, we prove that if $B_n$ denotes the number of minimal balanced collections, then $\frac{0.288}{n!} \, 2^{(n-1)^2} < B_n < \frac{120}{n!} \, 2^{n^2 - n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorics of Minimal Balanced Collections Bludov, Mikhail V. Zuev, Nikolai K. Combinatorics In this article, we explore the combinatorics of balanced collections. A collection of subsets of the set $[n] = \{1, \dots, n\}$ is called \emph{balanced} if the relative interior of the convex hull of the corresponding characteristic vectors intersects the main diagonal of the $n$-dimensional cube, and it is called \emph{minimal} if it contains no proper balanced subcollections. In particular, we establish both upper and lower bounds for the number of minimal balanced collections. Specifically, we prove that if $B_n$ denotes the number of minimal balanced collections, then $\frac{0.288}{n!} \, 2^{(n-1)^2} < B_n < \frac{120}{n!} \, 2^{n^2 - n}$. |
| title | Combinatorics of Minimal Balanced Collections |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.19323 |