Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2024; 61(3): 621-635

Online first article March 19, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230269

Copyright © The Korean Mathematical Society.

The unimodality of the $r_3$-crank of $3$-regular overpartitions

Robert XiaoJian Hao, Erin YiYing Shen

Nanjing Institute of Technology; Hohai University

Abstract

An $l$-regular overpartition of $n$ is an overpartition of $n$ with no parts divisible by $l$. Recently, the authors introduced a partition statistic called $r_l$-crank of $l$-regular overpartitions. Let $M_{r_l}(m,n)$ denote the number of $l$-regular overpartitions of $n$ with $r_l$-crank $m$. In this paper, we investigate the monotonicity property and the unimodality of $M_{r_3}(m,n)$. We prove that $M_{r_3}(m,n)\geq M_{r_3}(m,n-1)$ for any integers $m$ and $n \geq 6$ and the sequence $\{M_{r_3}(m,n)\}_{|m|\leq n}$ is unimodal for all $n\geq 14$.

Keywords: Regular overpartition, $r_l$-crank, monotonicity, unimodality

MSC numbers: Primary 05A17, 11P83, 05A20

Supported by: This work was financially supported by NSFC 12101307 and 11801139 and the Qing Lan Project of JiangSu Province.