In this paper, we study a class of partitions called unrestricted singular overpartitions with parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ k $</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ i $</EquationSource> <EquationSource Format="MATHML"><math> <mi>i</mi> </math></EquationSource> </InlineEquation>, which is an overpartition such that no part is divisible by k, and any single occurrence of a part congruent to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\pm i \pmod {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mi>i</mi> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> may be overlined. The number of all such overpartitions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ n $</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> is denoted by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\overline{Cu}_{k,i}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">Cu</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Previous studies have established several congruence properties for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\overline{Cu}_{3,1}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">Cu</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo 2 and 4, and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\overline{Cu}_{4,1}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">Cu</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mn>4</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo 2, 4, and 8. We extend these results by deriving new congruences for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\overline{Cu}_{6,1}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">Cu</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mn>6</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> modulo 36, 48, and 64. In addition, we establish connections between unrestricted singular overpartitions and other partition families, including <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(k,\ell )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-regular bipartitions and overpartitions with <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-regular nonoverlined parts.

Arithmetic properties for unrestricted singular overpartitions

期刊论文
发表日期：

2026年8月5日

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Yosua Feri Wijaya,
Uha Isnaini,
Yeni Susanti,
Pee Choon Toh

摘要

In this paper, we study a class of partitions called unrestricted singular overpartitions with parameters $ k $ $k$ and $ i $ $i$ , which is an overpartition such that no part is divisible by k, and any single occurrence of a part congruent to $\pm i \pmod {k}$ $\pm i (mod k)$ may be overlined. The number of all such overpartitions of $ n $ $n$ is denoted by ${\overline{Cu}_{k,i}}(n)$ ${\bar{Cu}}_{k, i} (n)$ . Previous studies have established several congruence properties for ${\overline{Cu}_{3,1}}(n)$ ${\bar{Cu}}_{3, 1} (n)$ modulo 2 and 4, and ${\overline{Cu}_{4,1}}(n)$ ${\bar{Cu}}_{4, 1} (n)$ modulo 2, 4, and 8. We extend these results by deriving new congruences for ${\overline{Cu}_{6,1}}(n)$ ${\bar{Cu}}_{6, 1} (n)$ modulo 36, 48, and 64. In addition, we establish connections between unrestricted singular overpartitions and other partition families, including $(k,\ell )$ $(k, ℓ)$ -regular bipartitions and overpartitions with $\ell $ $ℓ$ -regular nonoverlined parts.