Enumeration of M-partitions

An M-partition of a positive integer mm is a partition of mm with as few parts as possible such that every positive integer less than mm can be written as a sum of parts taken from the partition. This type of partition is a variation of MacMahonʹs perfect partition, and was recently introduced and studied by O’Shea, who showed that for half the numbers mm, the number of M-partitions of mm is equal to the number of binary partitions of 2n+1-1-m2n+1-1-m, where View the MathML sourcen=⌊log2m⌋. In this note we extend O’Sheaʹs result to cover all numbers mm.