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 m-ary 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.
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- Generating function