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arXiv · 2511.19323

Enumerating Minimal Balanced Collections

Mikhail V. Bludov·Nikolai K. Zuev
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Abstract

In this note, 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 at a point other than the origin, and it is called \emph{minimal} if it contains no proper balanced subcollections. We determine the asymptotic number of minimal balanced collections. Specifically, if $B_n$ denotes their total number, then $B_n=2^{n^2-n+1}\bigl(1+o(1)\bigr)/n!$ as $n\to\infty$. We discuss applications of this result to fractional matchings in hypergraphs and to resonance arrangement

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Mikhail V. Bludov, Nikolai K. Zuev. 2025-11-24. Enumerating Minimal Balanced Collections. https://arxiv.org/abs/2511.19323

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