A complete solution of the partition of a number into arithmetic progressions

Abstract

We solve the enumeration of the set $\textrm{AP}(n)$ of partitions of a positive integer $n$ in which the nondecreasing sequence of parts forms an arithmetic progression. In particular, we establish a formula for the number of nondecreasing arithmetic progressions of positive integers with sum $n$. We also present an explicit method to calculate all the partitions of $\textrm{AP}(n)$.