Recently, Amdeberhan and Merca introduced the function a(n) that counts the number of partitions of n whose odd parts may appear in one of three distinct colors and whose even parts appear in only one color. We employ the Atkin’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$U_7$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mn>7</mn> </msub> </math></EquationSource> </InlineEquation> operator on a certain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-quotient on the congruence subgroup <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma _0(98)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>98</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to find the generating function for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a(7n+2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> modulo 49. Consequently, we apply elementary q-series manipulations and the result of Ahlgren on this generating function to deduce that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a(343n+100)\equiv 0\pmod {49}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mn>343</mn> <mi>n</mi> <mo>+</mo> <mn>100</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>49</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$n\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Congruences modulo 49 for partitions with 3-colored odd parts

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发表日期：

2025年11月17日

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Russelle Guadalupe

摘要

Recently, Amdeberhan and Merca introduced the function a(n) that counts the number of partitions of n whose odd parts may appear in one of three distinct colors and whose even parts appear in only one color. We employ the Atkin’s $U_7$ $U_{7}$ operator on a certain $\eta $ $η$ -quotient on the congruence subgroup $\Gamma _0(98)$ $Γ_{0} (98)$ to find the generating function for $a(7n+2)$ $a (7 n + 2)$ modulo 49. Consequently, we apply elementary q-series manipulations and the result of Ahlgren on this generating function to deduce that $a(343n+100)\equiv 0\pmod {49}$ $a (343 n + 100) \equiv 0 (mod 49)$ for all $n\ge 0$ $n \geq 0$ .