<p>We consider the set of partitions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( ped (n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi>e</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which counts the number of partitions of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation> wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences modulo 192 which were conjectured by Nath [10]. Further, we prove a few infinite families of congruences modulo 24 by using a result of Newman. Also, we prove that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(ped (9n+7)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi>e</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mi>n</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is lacunary modulo <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2^{k+2}\cdot 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>·</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(3^{k+1}\cdot 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>3</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>·</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> for all positive integers <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(k\geq0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We further prove an infinite family of congruences for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(ped (n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi>e</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.</p>

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Congruences and density results for partitions into distinct even parts

  • H. Nath,
  • A. Sarma

摘要

We consider the set of partitions \( ped (n)\) p e d ( n ) which counts the number of partitions of \(n\) n wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences modulo 192 which were conjectured by Nath [10]. Further, we prove a few infinite families of congruences modulo 24 by using a result of Newman. Also, we prove that \(ped (9n+7)\) p e d ( 9 n + 7 ) is lacunary modulo \(2^{k+2}\cdot 3\) 2 k + 2 · 3 and \(3^{k+1}\cdot 4\) 3 k + 1 · 4 for all positive integers \(k\geq0\) k 0 . We further prove an infinite family of congruences for \(ped (n)\) p e d ( n ) modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.