2003/11/14 by Edwin O'Shea, Edwin O’Shea, O'Shea, Edwin
Mathematics · #05A17 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A17
paper · pdf · doi:10.48550/arxiv.math/0311230
11 pages
arxiv created 2003/11/14 · openalex publication_date 2003/11/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An M-partition of a positive integer m is a partition with as few parts as possible such that any positive integer less than m has a partition made up of parts taken from that partition of m. This is equivalent to partitioning a weight m so as to be able to weigh any integer weight l < m with as few weights as possible and only one scale pan. We show that the number of parts of an M-partition is a log-linear function of m and the M-partitions of m correspond to lattice points in a polytope. We exhibit a recurrence relation for counting the number of M-partitions of m and, for ``half'' of the positive integers, this recurrence relation will have a generating function. The generating function will be, in some sense, the same as the generating function for counting the number of distinct binary partitions for a given integer.