A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts
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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BibTeX
@misc{liu2026proof,
title = {{A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts}},
author = {Eric H. Liu and X. L. Liu and Olivia X. M. Yao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27350},
eprint = {2609.27350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27350},
}
RIS
TY - UNPB
TI - A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts
AU - Eric H. Liu
AU - X. L. Liu
AU - Olivia X. M. Yao
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27350
UR - https://arxiv.org/abs/2609.27350
AB - 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
ER -