发表机构
National Institute of Technology, Raipur（Raipur国家技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用统一矩-迹框架，对$t$-核分拆和超分拆的曲柄统计量建立迹与逆迹公式，并借助完全Bell多项式导出分拆数的显式表达式。
AI 中文摘要
最近，Kang、Kim 和 Lee 利用完全Bell多项式及其反演公式，为对称分拆统计量建立了一个统一的矩-迹框架。在本文中，我们将该框架应用于$t$-核分拆和超分拆的曲柄统计量。对于$t\in\{5,7,11,17,19\}$，我们证明了$t$-核分拆的归一化偶阶曲柄矩生成函数可表示为函数$D^{(t)}_{2s}(\tau)$的分拆迹形式，并带有适当的伯努利数平移。我们还建立了逆迹公式，从相应的归一化偶阶曲柄矩中恢复$D^{(t)}_{2s}(\tau)$。对于超分拆，我们获得了与第一和第二剩余曲柄生成函数相关的归一化偶阶矩的类似迹和逆迹恒等式。作为应用，我们使用完全Bell多项式及其反演公式，得到了$t$-核分拆数和超分拆数关于除数函数和的显式表达式。
英文摘要
Recently, Kang, Kim, and Lee \cite{Kang2026} developed a unified moment-trace framework for symmetric partition statistics using complete Bell polynomials and their inversion formula. In this paper, we apply this framework to crank statistics for $t$-core partitions and overpartitions. For $t\in\{5,7,11,17,19\}$, we show that the normalized even crank moment generating functions for $t$-core partitions admit partition-trace representations in terms of the functions $D^{(t)}_{2s}(τ)$, together with suitable Bernoulli-number shifts. We also establish inverse trace formulas that recover $D^{(t)}_{2s}(τ)$ from the corresponding normalized even crank moments. For overpartitions, we obtain analogous trace and inverse-trace identities for the normalized even moments associated with the first and second residual crank generating functions. As applications, we use complete Bell polynomials and their inversion formula to obtain explicit expressions for the $t$-core partition numbers and overpartitions number in terms of sums involving divisor function.
CommentsFirst draft of the paper. Any comments are welcome
