On Euler's Theorem
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918188143345664 |
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| author | Andrews, George E. Kumar, Rahul Yee, Ae Ja |
| author_facet | Andrews, George E. Kumar, Rahul Yee, Ae Ja |
| contents | Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*} A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1), \end{align*} where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_03979 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Euler's Theorem Andrews, George E. Kumar, Rahul Yee, Ae Ja Combinatorics Number Theory Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$, \begin{align*} A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1), \end{align*} where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated. |
| title | On Euler's Theorem |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2511.03979 |