Estimates on binomial sums of partition functions

TL;DR

This paper establishes bounds on binomial sums of partition functions, proving their unimodality and deriving upper bounds that have applications in estimating minimal dimensions of faithful modules for nilpotent Lie algebras.

Contribution

It introduces new bounds on the sums of partition functions, demonstrating their unimodality and providing applications to Lie algebra representation theory.

Findings

Proves $p(n,k)$ is unimodal.

Derives an upper bound $p(n,k) < rac{2.825}{ ext{sqrt}(n)} 2^n$ for fixed $n$.

Provides bounds for $p(n,n-1)$ and $p(n-1,n-1)$ related to exponential functions.

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 $p(n,n-1)<\sqrt{n}\exp(\pi\sqrt{2n/3})$ for $n\ge 1$ and $p(n-1,n-1)<\exp (\pi\sqrt{2n/3} )$.