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

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