Enumeration of M-partitions

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