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arXiv · 2611.08109

The Calkin--Wilf problems on $\ell$-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 $b_{k,\ell}(n)$, the number of partitions of $n$ into powers of $\ell$ 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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Michael Coons. 2026-11-05. The Calkin--Wilf problems on $\ell$-ary partition functions. https://arxiv.org/abs/2611.08109

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