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 \(\beta \).