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

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.