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PreprintCC BY 4.0arXiv

The Calkin--Wilf problems on ℓ-ary partition functions

  • Michael Coons
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Abstract

In their 1998 precursor to their celebrated paper ``Recounting the rationals,'' Calkin and Wilf listed nine open problems related to the function bk,ℓ(n), the number of partitions of n into powers of ℓ with each part appearing at most k times. In this short paper, we answer five of their eight remaining open problems. As a consequence of our results, we extend and generalise results of both Carlitz and Reznick, and answer an open question of Reznick.

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Cite

BibTeXRIS
BibTeX
@misc{coons2026calkinwilf,
  title = {{The Calkin--Wilf problems on $\ell$-ary partition functions}},
  author = {Michael Coons},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08109},
  eprint = {2611.08109},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08109},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Calkin--Wilf problems on ℓ-ary partition functions
AU  - Michael Coons
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08109
UR  - https://arxiv.org/abs/2611.08109
AB  - In their 1998 precursor to their celebrated paper ``Recounting the rationals,'' Calkin and Wilf listed nine open problems related to the function b_k,ℓ(n), the number of partitions of n into powers of ℓ with each part appearing at most k times. In this short paper, we answer five of their eight remaining open problems. As a consequence of our results, we extend and generalise results of both Carlitz and Reznick, and answer an open question of Reznick.
ER  -