Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2601.09472 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915729736990720 |
|---|---|
| author | Burde, Dietrich |
| author_facet | Burde, Dietrich |
| contents | Let $p(n)$ denote the partition function and define $p(n,k)=\sum_{j=0}^{k}\binom{n-j}{k-j}p(j)$ where $p(0)=1$. We prove that $p(n,k)$ is unimodal and satisfies $p(n,k) < \frac{2.825}{\sqrt{n}}\, 2^n $ for fixed $n\ge 1$ and all $1\le k\le n$. This result has an interesting application: the minimal dimension of a faithful module for a $k$-step nilpotent Lie algebra of dimension $n$ is bounded by $p(n,k)$ and hence by $\frac{3}{\sqrt{n}}\, 2^n $, independently of $k$. So far only the bound $n^{n-1}$ was known. We will also prove that $p(n,n-1)<\sqrt{n}\exp(π\sqrt{2n/3})$ for $n\ge 1$ and $p(n-1,n-1)<\exp (π\sqrt{2n/3} )$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09472 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Estimates on binomial sums of partition functions Burde, Dietrich Number Theory 11P81 Let $p(n)$ denote the partition function and define $p(n,k)=\sum_{j=0}^{k}\binom{n-j}{k-j}p(j)$ where $p(0)=1$. We prove that $p(n,k)$ is unimodal and satisfies $p(n,k) < \frac{2.825}{\sqrt{n}}\, 2^n $ for fixed $n\ge 1$ and all $1\le k\le n$. This result has an interesting application: the minimal dimension of a faithful module for a $k$-step nilpotent Lie algebra of dimension $n$ is bounded by $p(n,k)$ and hence by $\frac{3}{\sqrt{n}}\, 2^n $, independently of $k$. So far only the bound $n^{n-1}$ was known. We will also prove that $p(n,n-1)<\sqrt{n}\exp(π\sqrt{2n/3})$ for $n\ge 1$ and $p(n-1,n-1)<\exp (π\sqrt{2n/3} )$. |
| title | Estimates on binomial sums of partition functions |
| topic | Number Theory 11P81 |
| url | https://arxiv.org/abs/2601.09472 |