Summary: A minimal $r$-complete partition of an integer $m$ is a partition of $m$ with as few parts as possible, such that all the numbers 1,$\dots , rm$ can be written as a sum of parts taken from the partition, each part being used at most $r$ times. This is a generalization of M-partitions (minimal 1-complete partitions). The number of M-partitions of $m$ was recently connected to the binary partition function and two related arithmetic functions. In this paper we study the case $r \geq 2$, and connect the number of minimal $r$-complete partitions to the $(r+1)$-ary partition function and a related arithmetic function.

11P81, 05A17

complete partitions, M-partitions, (r + 1)-ary partitions