m-ary partitions with no gaps: A characterization modulo m

Abstract. In a recent work, the authors provided the first-ever characterization of the values bm(n) modulo m where bm(n) is the number of (unrestricted) m-ary partitions of the integer n and m ≥ 2 is a fixed integer. That characterization proved to be quite elegant and relied only on the base m representation of n. Since then, the authors have been motivated to consider a specific restricted m-ary partition function, namely cm(n), the number of mary partitions of n where there are no “gaps” in the parts. (That is to say, if mi is a part in a partition counted by cm(n), and i is a positive integer, then mi−1 must also be a part in the partition.) Using tools similar to those utilized in the aforementioned work on bm(n), we prove the first-ever characterization of cm(n) modulo m. As with the work related to bm(n) modulo m, this characterization of cm(n) modulo m is also based solely on the base m representation of n. 2010 Mathematics Subject Classification: 05A17, 11P83

• Type: Journal
• Year: 2016
• Where: DM
• Authors: George E. Andrews, Aviezri S. Fraenkel, James A. Sellers
• Keywords: DM 2016