Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B_{u,v}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of (u, v)-regular bipartitions of n. In this article, we prove that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$B_{p,m}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is always almost divisible by p,  where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> is a prime number and m is a positive odd integer with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gcd (p,3m)=1. $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mn>3</mn> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Further, we obtain infinite families of congruences modulo 3 for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B_{3,7}(n),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$B_{3,5}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$B_{3,2}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>3</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by using the theory of Hecke eigenforms and a result of Newman (Ann Math 70:478–489, 1959). Furthermore, we get many infinite families of congruences modulo 7, 11, and 13 for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$B_{2,7}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>7</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$B_{2,11}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>11</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$B_{2,13}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>13</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, respectively, by employing an identity of Newman (Ann Math 70:478–489, 1959). In addition, we prove infinite families of congruences modulo 2 for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$B_{4,3}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>4</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$B_{8,3}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>8</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$B_{4,5}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mrow> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by applying another result of Newman (Ann Math 75:242–250, 1962).

Distribution and congruences of (u, v)-regular bipartitions

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发表日期：

2026年3月7日

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Nabin Kumar Meher

摘要

Let $B_{u,v}(n)$ $B_{u, v} (n)$ denote the number of (u, v)-regular bipartitions of n. In this article, we prove that $B_{p,m}(n)$ $B_{p, m} (n)$ is always almost divisible by p, where $p\ge 5$ $p \geq 5$ is a prime number and m is a positive odd integer with $\gcd (p,3m)=1. $ $gcd (p, 3 m) = 1 .$ Further, we obtain infinite families of congruences modulo 3 for $B_{3,7}(n),$ $B_{3, 7} (n),$ $B_{3,5}(n)$ $B_{3, 5} (n)$ and $B_{3,2}(n)$ $B_{3, 2} (n)$ by using the theory of Hecke eigenforms and a result of Newman (Ann Math 70:478–489, 1959). Furthermore, we get many infinite families of congruences modulo 7, 11, and 13 for $B_{2,7}(n)$ $B_{2, 7} (n)$ , $B_{2,11}(n)$ $B_{2, 11} (n)$ , and $B_{2,13}(n)$ $B_{2, 13} (n)$ , respectively, by employing an identity of Newman (Ann Math 70:478–489, 1959). In addition, we prove infinite families of congruences modulo 2 for $B_{4,3}(n)$ $B_{4, 3} (n)$ , $B_{8,3}(n)$ $B_{8, 3} (n)$ , and $B_{4,5}(n)$ $B_{4, 5} (n)$ by applying another result of Newman (Ann Math 75:242–250, 1962).