## 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 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.

Original language | English (US) |
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Article number | 10238 |

Pages (from-to) | 283-287 |

Number of pages | 5 |

Journal | Discrete Mathematics |

Volume | 339 |

Issue number | 1 |

DOIs | |

State | Published - Jan 6 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015 Elsevier B.V.

## Keywords

- Congruence
- Generating function
- Partition