Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{M2spt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>M2spt</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the number of smallest parts in the partitions of n without repeated odd parts and with smallest part even. Ahlgren, Bringmann and Lovejoy, and Garvan and Jennings-Shaffer proved some congruences modulo <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p^m$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>p</mi> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{M2spt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>M2spt</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> is a prime and m is a positive integer. However, there is not as much known for congruences modulo 2 for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{M2spt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>M2spt</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we establish a parity relation between <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{M2spt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>M2spt</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the Hurwitz class number. In addition, we prove an infinite families of congruences modulo 2 for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{M2spt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>M2spt</mtext> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Parity results on the number of occurrences of the smallest parts in the partitions without repeated odd parts and with smallest part even

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发表日期：

2025年11月17日

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Jing Jin,
Eric H. Liu,
Ernest X. W. Xia

摘要

Let $\textrm{M2spt}(n)$ $M2spt (n)$ denote the number of smallest parts in the partitions of n without repeated odd parts and with smallest part even. Ahlgren, Bringmann and Lovejoy, and Garvan and Jennings-Shaffer proved some congruences modulo $p^m$ $p^{m}$ for $\textrm{M2spt}(n)$ $M2spt (n)$ , where $p\ge 3$ $p \geq 3$ is a prime and m is a positive integer. However, there is not as much known for congruences modulo 2 for $\textrm{M2spt}(n)$ $M2spt (n)$ . In this paper, we establish a parity relation between $\textrm{M2spt}(n)$ $M2spt (n)$ and the Hurwitz class number. In addition, we prove an infinite families of congruences modulo 2 for $\textrm{M2spt}(n)$ $M2spt (n)$ .