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arXiv · 2609.27350

A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts

Eric H. Liu·X. L. Liu·Olivia X. M. Yao
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Abstract

Andrews and Dastidar introduced the statistic $\SOME(n)$, defined as the difference between the sum of all odd parts and the sum of all even parts occurring in the partitions of $n$, and investigated its arithmetic properties. Motivated by their work, Baruah and Gogoi defined two analogous statistics for overpartitions, namely $\OSOMEo(n)$ and $\OSOMEe(n)$, which record the sums of all odd and all even overlined parts, respectively. They established several congruences for these statistics and their difference, and proposed a conjecture on congruences modulo $7$. In this paper, we prove their conjecture by elementary $q$-series methods. Our proof combines two specializations of Watson's quintuple product identity, a $(p,k)$-parameterization of divisor-sum series, and termwise logarithmic differentiation

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Eric H. Liu, X. L. Liu, Olivia X. M. Yao. 2026-09-23. A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts. https://arxiv.org/abs/2609.27350

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