Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{po}_k(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">po</mi> </mrow> <mo>¯</mo> </mover> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of k-colored overpartitions of n into odd parts. By deriving the exact generating functions for specific sets of arithmetic progressions in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{po}_{2k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">po</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we prove several infinite families of congruences modulo high powers of 2 satisfied by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\overline{po}_{2k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">po</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This significantly generalizes some results of Adiga and Ranganatha (Discrete Math 341(11):3141–3147, 2018). Furthermore, we propose a conjecture concerning a family of congruences modulo high powers of 2 for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\overline{po}_{2^{k}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mrow> <mi mathvariant="italic">po</mi> </mrow> <mo>¯</mo> </mover> <msup> <mn>2</mn> <mi>k</mi> </msup> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

Congruences modulo powers of 2 for k-colored overpartitions into odd parts

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发表日期：

2026年4月16日

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Dazhao Tang

摘要

Let $\overline{po}_k(n)$ ${\bar{po}}_{k} (n)$ denote the number of k-colored overpartitions of n into odd parts. By deriving the exact generating functions for specific sets of arithmetic progressions in $\overline{po}_{2k}(n)$ ${\bar{po}}_{2 k} (n)$ , we prove several infinite families of congruences modulo high powers of 2 satisfied by $\overline{po}_{2k}(n)$ ${\bar{po}}_{2 k} (n)$ . This significantly generalizes some results of Adiga and Ranganatha (Discrete Math 341(11):3141–3147, 2018). Furthermore, we propose a conjecture concerning a family of congruences modulo high powers of 2 for $\overline{po}_{2^{k}}(n)$ ${\bar{po}}_{2^{k}} (n)$ when $k\ge 3$ $k \geq 3$ .