Bull. Korean Math. Soc.
Online first article March 19, 2024
Copyright © The Korean Mathematical Society.
Robert X. J. Hao and Erin Y. Y. Shen
Nanjing Institute of Technology, Hohai University
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: 05A17, 11P83, 05A20
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd