Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b_{t,k}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of hooks of length k in all the t-regular partitions of n. Recently, the first and the third authors proved that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b_{3,2}(n)\ge b_{2,2}(n)$</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> <mo>≥</mo> <msub> <mi>b</mi> <mrow> <mn>2</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> for all <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and conjectured that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$b_{t+1,2}(n)\ge b_{t,2}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msub> <mi>b</mi> <mrow> <mi>t</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove that the conjecture is true for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$t=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.

Hook length inequalities for t-regular partitions in the t-aspects: Part II

期刊论文
发表日期：

2025年9月5日

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Rupam Barman,
Pankaj Jyoti Mahanta,
Gurinder Singh

摘要

Let $t\ge 2$ $t \geq 2$ and $k\ge 1$ $k \geq 1$ be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let $b_{t,k}(n)$ $b_{t, k} (n)$ denote the number of hooks of length k in all the t-regular partitions of n. Recently, the first and the third authors proved that $b_{3,2}(n)\ge b_{2,2}(n)$ $b_{3, 2} (n) \geq b_{2, 2} (n)$ for all $n\ge 4$ $n \geq 4$ , and conjectured that $b_{t+1,2}(n)\ge b_{t,2}(n)$ $b_{t + 1, 2} (n) \geq b_{t, 2} (n)$ for all $t\ge 3$ $t \geq 3$ and $n\ge 0$ $n \geq 0$ . In this paper, we prove that the conjecture is true for $t=3$ $t = 3$ .