Elsevier

Discrete Mathematics

Volume 339, Issue 1, 6 January 2016, Pages 283-287
Discrete Mathematics

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

https://doi.org/10.1016/j.disc.2015.08.016Get rights and content
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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 m2 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 mi1 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.

Keywords

Partition
Congruence
Generating function

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