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Difference of the Sum of All Odd and Even Overlined Parts of Overpartitions
Abstract
Recently, Garvan and Sarma studied sums of non-overlined parts in overpartitions and partitions without repeated odd parts. In this paper, we study the corresponding statistic for overlined parts. We define OSOME(n) as the sum of all odd overlined parts minus the sum of all even overlined parts in the overpartitions of n. We obtain a closed form for its generating function and use it to prove congruences by elementary q-series methods, the theory of half-integral weight modular forms, and known congruences for the overpartition function.
Related
- A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts
- A Combinatorial Proof of an Identity of Malik and Sarma on Overpartitions with Repeated Smallest Non-Overlined Part
- Internal congruences modulo powers of 2 for k-colored overpartitions into odd parts
- On partitions with bounded multiplicity restrictions on even parts and with the largest part being odd
- Asymptotic Formula for Biregular Overpartitions
BibTeX
@misc{baruah2026difference,
title = {{Difference of the Sum of All Odd and Even Overlined Parts of Overpartitions}},
author = {Nayandeep Deka Baruah and Pankaj Gogoi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21361},
eprint = {2609.21361},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21361},
}
RIS
TY - UNPB
TI - Difference of the Sum of All Odd and Even Overlined Parts of Overpartitions
AU - Nayandeep Deka Baruah
AU - Pankaj Gogoi
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21361
UR - https://arxiv.org/abs/2609.21361
AB - Recently, Garvan and Sarma studied sums of non-overlined parts in overpartitions and partitions without repeated odd parts. In this paper, we study the corresponding statistic for overlined parts. We define OSOME(n) as the sum of all odd overlined parts minus the sum of all even overlined parts in the overpartitions of n. We obtain a closed form for its generating function and use it to prove congruences by elementary q-series methods, the theory of half-integral weight modular forms, and known congruences for the overpartition function.
ER -