Moments of the Crank Statistic for $t$-Core Partitions and Overpartitions

RESUMO

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 $D^{(t)}_{2s}(\tau)$, together with suitable Bernoulli-number shifts. We also establish inverse trace formulas that recover $D^{(t)}_{2s}(\tau)$ 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.