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arXiv · 2608.05142

A bijective proof of a partition theorem of Berkovich and Uncu

Michal Mogielnicki·Ken Ono·Niels Voss·Jujian Zhang
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Abstract

In 2016, Berkovich and Uncu proved that, for all nonnegative integers $i$, $j$, and $n$, the number of strict partitions of $n$ with $i$ odd-indexed odd parts and $j$ even-indexed odd parts equals the number of strict partitions of $n$ with $i$ parts congruent to $1$ modulo $4$ and $j$ parts congruent to $3$ modulo $4$. Their proof used generating functions, and they asked for a combinatorial proof. We answer their question with an explicit bijection, assembled from three classical ingredients: $2$-modular diagrams, an insertion algorithm of Chen, Gao, Ji, and Li, and Glaisher's bijection. AxiomProver autonomously formalized and verified the proof of the main theorem in Lean.

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Michal Mogielnicki, Ken Ono, Niels Voss, Jujian Zhang. 2026-08-05. A bijective proof of a partition theorem of Berkovich and Uncu. https://arxiv.org/abs/2608.05142

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