<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\widetilde{\operatorname {pe}}(\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mo>pe</mo> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the restricted parity excess, which is the number of different odd parts minus the number of different even parts in the partition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. We define a random variable <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$X_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> which takes value <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\widetilde{\operatorname {pe}}(\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mo>pe</mo> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> while we choose <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> uniformly at random from the set of partitions of <i>n</i>. Then, we show that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$X_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is asymptotically normally distributed.</p>

On the distribution of a restricted parity excess

期刊论文
发表日期：

2026年4月24日

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Byungchan Kim,
Eunmi Kim

摘要

Let $\widetilde{\operatorname {pe}}(\lambda )$ $\tilde{pe} (λ)$ be the restricted parity excess, which is the number of different odd parts minus the number of different even parts in the partition $\lambda $ $λ$ . We define a random variable $X_n$ $X_{n}$ which takes value $\widetilde{\operatorname {pe}}(\lambda )$ $\tilde{pe} (λ)$ while we choose $\lambda $ $λ$ uniformly at random from the set of partitions of n. Then, we show that $X_n$ $X_{n}$ is asymptotically normally distributed.