Integer partitions with restricted odd and even parts

Abstract

In this note, two generalized partition functions $p_o^\alpha (n)$ and $p_e^\beta (n)$ are considered, where for any odd positive integer $\alpha$, $p_o^\alpha (n)$ denotes the number of partitions of n into odd parts such that no parts is congruent to $\alpha$ modulo $2\alpha$, and for any even positive integer $\beta$, $p_e^\beta (n)$ denotes the number of partitions of n into even parts such that no parts is congruent to $\beta$ modulo $2\beta$. Some divisibility properties of $p_o^\alpha (n)$ and $p_e^\beta (n)$ are discussed for some particular values of $\alpha$ and $\beta$.