In this paper, we study the restricted partition function pod(n), which counts the number of partitions of n with distinct odd parts and unrestricted even parts. Building on a Rademacher-type formula established by Sills, we derive sharp bounds for pod(n) through precise truncations and error analysis of the convergent series. These bounds enable us to prove several important inequalities for large n. In particular, we show that pod(n) satisfies the Turán inequality for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n \ge 47$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>47</mn> </mrow> </math></EquationSource> </InlineEquation>, the higher-order Turán inequality for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 139$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>139</mn> </mrow> </math></EquationSource> </InlineEquation>, and Laguerre-type inequalities of order 2 and 3 for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \ge 261$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>261</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n \ge 728$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>728</mn> </mrow> </math></EquationSource> </InlineEquation>, respectively. In addition, we establish the determinantal inequalities of order 3 for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n \ge 309$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>309</mn> </mrow> </math></EquationSource> </InlineEquation>, paralleling known results for the classical partition function.

Inequalities for partitions with odd parts distinct

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发表日期：

2025年12月9日

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Edward Y. S. Liu,
Hongtao Xiang,
Jane Y. X. Yang

摘要

In this paper, we study the restricted partition function pod(n), which counts the number of partitions of n with distinct odd parts and unrestricted even parts. Building on a Rademacher-type formula established by Sills, we derive sharp bounds for pod(n) through precise truncations and error analysis of the convergent series. These bounds enable us to prove several important inequalities for large n. In particular, we show that pod(n) satisfies the Turán inequality for $n \ge 47$ $n \geq 47$ , the higher-order Turán inequality for $n \ge 139$ $n \geq 139$ , and Laguerre-type inequalities of order 2 and 3 for $n \ge 261$ $n \geq 261$ and $n \ge 728$ $n \geq 728$ , respectively. In addition, we establish the determinantal inequalities of order 3 for $n \ge 309$ $n \geq 309$ , paralleling known results for the classical partition function.