Moments of the Crank Statistic for t-Core Partitions and Overpartitions
Abstract
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∈{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.
Related
- The Algebra of Parity-Twisted Crank Moments and a Prime-Detecting Expression
- Sharp Euler-factor positivity for the BG-rank-refined five-core crank
- On the Moments of Least s-Gaps of Partitions and a Conjecture of Baruah and Talukdar
- Maximum length and size of simultaneous core partitions with distinct parts
- A Prime-Power Dichotomy for Nekrasov--Okounkov Hook Lengths and t-Core Partitions
BibTeX
@misc{bhowmick2026moments,
title = {{Moments of the Crank Statistic for $t$-Core Partitions and Overpartitions}},
author = {Sourav Bhowmick and Nabin Kumar Meher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08360},
eprint = {2609.08360},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08360},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Moments of the Crank Statistic for t-Core Partitions and Overpartitions
AU - Sourav Bhowmick
AU - Nabin Kumar Meher
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08360
UR - https://arxiv.org/abs/2609.08360
AB - 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∈{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.
ER -