PreprintCC BY 4.0arXiv
A Note On Certain Minimal Excludants Over Overpartitions
Abstract
Let σMex(n) and σeMex(n) denote the sum of minimal excludants and sum of even minimal excludants over all overpartitions, respectively. In this work, we study these two combinatorial objects from an arithmetic perspective. We prove that, for n≥ 0, σeMex(n)≡σMex(n)−p̅≥ 2(n)pmod22, where p̅≥ 2(n) denotes the number of overpartitions of n with all parts at least $2$, and obtain relation with basic hypergeometric series. Furthermore, we prove the asymptotic behavior of σMex(n), σeMex(n), and p̅≥ 2(n) as n→∞. In particular, we prove that σMex(n)∼ 2 σeMex(n).
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BibTeX
@misc{sarkar2026note,
title = {{A Note On Certain Minimal Excludants Over Overpartitions}},
author = {Dipika Sarkar and M. P. Thejitha and S. N. Fathima},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21649},
eprint = {2609.21649},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21649},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Note On Certain Minimal Excludants Over Overpartitions
AU - Dipika Sarkar
AU - M. P. Thejitha
AU - S. N. Fathima
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.21649
UR - https://arxiv.org/abs/2609.21649
AB - Let σMex(n) and σ_eMex(n) denote the sum of minimal excludants and sum of even minimal excludants over all overpartitions, respectively. In this work, we study these two combinatorial objects from an arithmetic perspective. We prove that, for n≥ 0, σ_eMex(n)≡σMex(n)−p̅_≥ 2(n)pmod2^2, where p̅_≥ 2(n) denotes the number of overpartitions of n with all parts at least $2$, and obtain relation with basic hypergeometric series. Furthermore, we prove the asymptotic behavior of σMex(n), σ_eMex(n), and p̅_≥ 2(n) as n→∞. In particular, we prove that σMex(n)∼ 2 σ_eMex(n).
ER -