On the sum of odd minimal excludants over overpartitions
Abstract
Andrews and Newman introduced the minimal excludant mex(λ) of an integer partition λ and studied the summatory function σmex(n), and Baruah et al. refined this to the odd- and even-restricted functions σomex(n) and σemex(n). In this paper, we introduce and study the overpartition analogue σomex̅(n), defined as the sum of odd minimal excludants over all overpartitions of n. We first derive the exact generating function for σomex̅(n), and relate it to the bivariate generating function of Aricheta and Donato for the overpartition minimal excludant. Using elementary q-series arguments, together with a weight-one eta-quotient identity for φ(q)2 verified via the Gordon--Hughes--Newman--Ligozat criterion, we establish an infinite family of congruences satisfied by σomex̅(n). Consequently, we obtain that σomex̅(0)=1 and, for n≥1, σomex̅(n) ≡ 0 pmod4 if and only if n is a perfect square. We further obtain congruences for the partial sums and self-convolution of σomex̅(n); in particular, an infinite family of congruences modulo $8$ for the self-convolution, expressed in terms of the divisor functions d1(n) and d3(n). We conclude the paper by establishing a Hardy--Ramanujan-type asymptotic formula for σomex̅(n) via the Wright circle method.
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BibTeX
@misc{v2026minimal,
title = {{On the sum of odd minimal excludants over overpartitions}},
author = {Veena V S and S N Fathima},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32207},
eprint = {2609.32207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32207},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the sum of odd minimal excludants over overpartitions
AU - Veena V S
AU - S N Fathima
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32207
UR - https://arxiv.org/abs/2609.32207
AB - Andrews and Newman introduced the minimal excludant mex(λ) of an integer partition λ and studied the summatory function σmex(n), and Baruah et al. refined this to the odd- and even-restricted functions σ_omex(n) and σ_emex(n). In this paper, we introduce and study the overpartition analogue σ_omex̅(n), defined as the sum of odd minimal excludants over all overpartitions of n. We first derive the exact generating function for σ_omex̅(n), and relate it to the bivariate generating function of Aricheta and Donato for the overpartition minimal excludant. Using elementary q-series arguments, together with a weight-one eta-quotient identity for φ(q)^2 verified via the Gordon--Hughes--Newman--Ligozat criterion, we establish an infinite family of congruences satisfied by σ_omex̅(n). Consequently, we obtain that σ_omex̅(0)=1 and, for n≥1, σ_omex̅(n) ≡ 0 pmod4 if and only if n is a perfect square. We further obtain congruences for the partial sums and self-convolution of σ_omex̅(n); in particular, an infinite family of congruences modulo $8$ for the self-convolution, expressed in terms of the divisor functions d_1(n) and d_3(n). We conclude the paper by establishing a Hardy--Ramanujan-type asymptotic formula for σ_omex̅(n) via the Wright circle method.
ER -