Unimodality of $k$-Regular Partitions into Distinct Parts with Bounded Largest Part

Abstract

A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the largest part is at most $km+k-1$. Let $d_{k,m}(n)$ denote the number of $k$-regular partition of $n$ into distinct parts where the largest part is at most $km+k-1$. In line with this method, we show that $d_{4,m}(n)\ge d_{4,m}(n-1)$ for $m\ge 0$, $1\le n\le 3(m+1{)}^2$ and $n\ne 4$ and $d_{8,m}(n)\ge d_{8,m}(n-1)$ for $m\ge 2$ and $1\le n\le 14(m+1{)}^2$. When $5\le k\le 10$ and $k\ne 8$, we show that $d_{k,m}(n)\ge d_{k,m}(n-1)$ for $m\ge 0$ and $1\le n\le \lfloor \frac{k(k-1)(m+1{)}^2}{4}\rfloor$.

19 pages