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arXiv · 2509.22618

The Number of Parts in the (Distinct) Partitions With Parts From a Set

A. David Christopher
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Abstract

The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the number of prime divisors of $n$, $p$-adic valuation of $n$, the number of Carlitz-binary compositions of $n$ and the Hamming weight function. Finally, we obtain an asymptotic estimate for the number of parts in the partitions of $n$ with parts from a finite set of relatively prime integers.

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A. David Christopher. 2025-09-26. The Number of Parts in the (Distinct) Partitions With Parts From a Set. https://arxiv.org/abs/2509.22618

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