Integer partitions with restricted odd and even parts

期刊论文
发表日期：

2024年4月17日

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Nipen Saikia

摘要

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