<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p_{-2,\ell }(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of 3-color partitions of <i>n</i> where one color appears solely in parts that are multiples of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ell \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>. This paper proves the existence of infinite families of congruences modulo powers of 5 for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p_{-2,\ell }(n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mrow> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>
Family of congruences for 3-color partition modulo powers of 5
Let \(p_{-2,\ell }(n)\) denote the number of 3-color partitions of n where one color appears solely in parts that are multiples of \(\ell \). This paper proves the existence of infinite families of congruences modulo powers of 5 for \(p_{-2,\ell }(n)\).
