The parity of the number of missing integers in partitions
Abstract
Missing integers in partitions have been studied by many authors. A missing integer of a partition is a positive integer smaller than its largest part that does not occur as a part. We investigate parity biases for the number of missing integers in partitions and overpartitions. For every integer c≥2, we prove that the coefficients ac(n) of (−cq;q)∞/(−q;q)∞ are eventually strictly positive and provide an effectively computable threshold. The proof uses the circle method, combining a saddle-point analysis on the major arc with explicit uniform estimates on the minor arcs. Specializing to c=2 and c=3, and verifying the remaining finite cases, we show that partitions and overpartitions of n with an even number of missing integers outnumber those with an odd number for n>34 and n>27, respectively. We further prove that, for n>10, the difference between the numbers of ordinary partitions with an even and an odd number of missing integers is smaller than the number of partitions of n into distinct parts. These results confirm three conjectures of Bhoria, Eyyunni and Santra. We also give a combinatorial interpretation of ac(n) for every integer c≥4 in terms of colored partitions, generalizing the partition and overpartition interpretations of Bhoria, Eyyunni and Santra for c=2 and c=3, respectively.
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BibTeX
@misc{du2026parity,
title = {{The parity of the number of missing integers in partitions}},
author = {Julia Q. D. Du and Gurinder Singh},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07908},
eprint = {2611.07908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07908},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The parity of the number of missing integers in partitions
AU - Julia Q. D. Du
AU - Gurinder Singh
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07908
UR - https://arxiv.org/abs/2611.07908
AB - Missing integers in partitions have been studied by many authors. A missing integer of a partition is a positive integer smaller than its largest part that does not occur as a part. We investigate parity biases for the number of missing integers in partitions and overpartitions. For every integer c≥2, we prove that the coefficients a_c(n) of (−cq;q)_∞/(−q;q)_∞ are eventually strictly positive and provide an effectively computable threshold. The proof uses the circle method, combining a saddle-point analysis on the major arc with explicit uniform estimates on the minor arcs. Specializing to c=2 and c=3, and verifying the remaining finite cases, we show that partitions and overpartitions of n with an even number of missing integers outnumber those with an odd number for n>34 and n>27, respectively. We further prove that, for n>10, the difference between the numbers of ordinary partitions with an even and an odd number of missing integers is smaller than the number of partitions of n into distinct parts. These results confirm three conjectures of Bhoria, Eyyunni and Santra. We also give a combinatorial interpretation of a_c(n) for every integer c≥4 in terms of colored partitions, generalizing the partition and overpartition interpretations of Bhoria, Eyyunni and Santra for c=2 and c=3, respectively.
ER -