Estimates on binomial sums of partition functions
Dietrich Burde
TL;DR
The paper studies the binomial-convolution sums $p(n,k)=\sum_{j=0}^{k} \binom{n-j}{k-j} p(j)$ with $p(n)$ the partition function, establishing unimodality in $k$ for fixed $n$ and a sharp upper bound $p(n,k) < (2.825/\sqrt{n}) 2^n$ for all $1\le k\le n$. These results yield improved bounds on the minimal dimension of a faithful module for nilpotent Lie algebras: $μ(\mathfrak{g})\le p(n,k)$, hence $μ(\mathfrak{g})\le (3/\sqrt{n}) 2^n$, refining earlier bounds such as $n^{n-1}$. In the extremal filiform case ($k=n-1$) the authors prove $p(n-1,n-1) < e^{\alpha\sqrt{n}}$ and $p(n,n-1) < \sqrt{n} e^{\alpha\sqrt{n}}$ with $\alpha=\sqrt{2/3}\pi$, and develop generating-function techniques tied to the Dedekind eta-function to obtain these and related bounds. The methods combine unimodality criteria for binomial convolutions with precise partition-function estimates, yielding both structural and quantitative improvements with implications for Ado-type bounds in Lie theory.
Abstract
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 $p(n-1,n-1)<\exp (π\sqrt{2n/3} )$.