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arXiv · 2509.17193

On asymptotic formula of the partition function $p_A(n)$

David Christopher·Davamani Christober
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Abstract

The partition function, $p_A(n)$, is defined to be the number of partitions of $n$ with parts in the set A, where $n$ is a positive integer and $A$ is a set of positive integers. It is well documented that: if A is a finite set with $\gcd(A)=1$ and $|A|=k$, then \[p_A(n)\sim \frac{n^{k-1}}{(\prod_{a\in A}a)(k-1)!}. \] Number of proofs have been obtained for this estimate. In this article, we give a new proof for the above estimate by making use of the fact that: $p_A(n)$ is a $quasi\ polynomial$ when A is a finite set. Present method of proof is purely combinatorial.

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David Christopher, Davamani Christober. 2025-09-21. On asymptotic formula of the partition function $p_A(n)$. https://arxiv.org/abs/2509.17193

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