On asymptotic formula of the partition function $p_A(n)$
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| Format: | Preprint |
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2025
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| _version_ | 1866915506114527232 |
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| author | Christopher, David Christober, Davamani |
| author_facet | Christopher, David Christober, Davamani |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17193 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On asymptotic formula of the partition function $p_A(n)$ Christopher, David Christober, Davamani Combinatorics Number Theory 11P82, 11P83 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. |
| title | On asymptotic formula of the partition function $p_A(n)$ |
| topic | Combinatorics Number Theory 11P82, 11P83 |
| url | https://arxiv.org/abs/2509.17193 |