Sourav Bhowmick | Searcharxiv

arXiv2026

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

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.

math.NT↗

arXiv2025

New Congruences on Biregular Overpartitions

Recently, Nadji, Ahmia and Ram\'{i}rez \cite{Nadji2025} investigated the arithmetic properties of ${\bar B}_{\ell_1,\ell_2}(n)$, the number of overpartitions where no part is divisible by $\ell_1$ or $\ell_2$ with $\gcd(\ell_1,\ell_2)$$=1$ and $\ell_1$, $\ell_2>1$. Specifically, they established congruences modulo $3$ and powers of $2$ for the pairs $(\ell_1, \ell_2)$ $\in$ $\{(4,3),(4,9),(8,3),(8,9)\}$, using the concept of generating functions, dissection formulas and Smoot's implementation of Radu's Ramanujan-Kolberg algorithm. Further, Alanazi, Munagi and Saikia \cite{Alanazi2024} established some congruences for the pairs $(\ell_1,\ell_2)$ $\in$ $\{(2,3),(4,3),(2,5),(3,5),(4,9),(8,27),(16,81)\}$ using the theory of modular forms and Radu's algorithm. Recently, Paudel, Sellers and Wang \cite{Paudel2025} extended several of their results and established infinitely many families of new congruences. In this paper, we find infinitely many families of congruences modulo $3$ and powers of $2$ for the pairs $(\ell_1,\ell_2)$ $\in$ $ \{(5,2^t), (4,3^t)\}$ $\forall t\geq1$ with $t\in\mathbb{N}$ and for $(3,2^t)$ $\forall t\geq2 $ with $t\in\mathbb{N}$, using the theory of Hecke eigenforms, an identity due to Newman \cite{Newman1959}, the concept of dissection formulas.

math.NT↗