Year: 2026
Type: preprint
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
Source: arXiv (Cornell University)
Authors: Liu, Eric H., Liu, X. L., Yao, Olivia X. M.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214201946