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Uniform Asymptotics for Andrews-Beck Partition Statistics
Abstract
Let NT(m,n) be the total number of parts in the partitions of n with rank m, and let MW(m,n) be the total number of ones in the partitions of n with crank m. Using the uniform generalized-rank estimate of Liu and Zhou, we establish asymptotic formulas for NT(m,n) and MW(m,n). Put βn=π/√(6n) and Ln=log(1/βn). For every fixed A>0, uniformly for |m|≤ A√n, we prove NT(m,n)&=p(n)Ln/4sech2 (βnm/2) +OA (p(n)log Ln) intertext and MW(m,n)&=p(n)/4log(1+e−βnm) sech2 (βnm/2) +OA (p(n)√(βnLn)).
Related
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BibTeX
@misc{mao2026uniform,
title = {{Uniform Asymptotics for Andrews-Beck Partition Statistics}},
author = {Renrong Mao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16024},
eprint = {2610.16024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16024},
}
RIS
TY - UNPB
TI - Uniform Asymptotics for Andrews-Beck Partition Statistics
AU - Renrong Mao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.16024
UR - https://arxiv.org/abs/2610.16024
AB - Let NT(m,n) be the total number of parts in the partitions of n with rank m, and let MW(m,n) be the total number of ones in the partitions of n with crank m. Using the uniform generalized-rank estimate of Liu and Zhou, we establish asymptotic formulas for NT(m,n) and MW(m,n). Put β_n=π/√(6n) and L_n=log(1/β_n). For every fixed A>0, uniformly for |m|≤ A√n, we prove NT(m,n)&=p(n)L_n/4sech^2 (β_nm/2) +O_A (p(n)log L_n) intertext and MW(m,n)&=p(n)/4log(1+e^−β_nm) sech^2 (β_nm/2) +O_A (p(n)√(β_nL_n)).
ER -