Chilean mathematician Nicolás Sanhueza-Matamala gave me the following problem:

Problem 1. Prove that for any \(c \leq n/3\) we have: \begin{equation} \sum_{r=1}^c \frac{c^r r^c}{r^{2r}} n^r = O(n^c). \end{equation}

Given that there are two variables involved, \(n\) and \(c\), to avoid confusion (some people were confused by this), let me restate the problem unrolling the big-Oh notation to avoid any potential ambiguities.

Problem 1. Define the function \(F(n, c) = \sum_{r = 1}^c \frac{c^r r^c}{r^{2r}} n^r\). Now prove that there is an absolute constant \(K\) and a natural \(n_0\) such that, for any \(n \geq n_0\) it holds that for every \(c \leq \frac{n}{3}\) we have \begin{equation} F(n, c) \leq K \cdot n^c. \end{equation}

This problem came up in Nicolás’ research in combinatorics, while trying to bound some combinatorial expression. At the end, this particular sub-problem didn’t end up being useful for his project, and thus instead of simply throwing this solution to the trash can, I have decided to put it up here.

My solution.

I now present my solution. If any reader of this blog has a nicer one, I’d be thrilled to check it out! Funnily enough, it is crucial that \(c \leq n/3\), as \(c \leq n/2.5\) would make this proof fail!

Start with the following algebra:

\[\begin{align} \sum_{r = 1}^c \frac{c^r r^c}{r^{2r}} n^r &= \sum_{k = 0}^{c-1} \frac{c^{(c-k)} (c-k)^c}{(c-k)^{2(c-k)}} n^{(c-k)}\\ &= \sum_{k = 0}^{c-1} \left(\frac{c}{c-k}\right)^{c-k} \cdot (c-k)^{k} n^{(c-k)}\\ &= \sum_{k = 0}^{c-1} \left(\frac{c}{c-k}\right)^{c-k} \cdot \left(\frac{c-k}{n}\right)^{k} n^{c}\\ &\leq \sum_{k = 0}^{c-1} \left(\frac{c}{c-k}\right)^{c-k} \cdot \left(\frac{c}{n}\right)^{k} n^{c}\\ &\leq \sum_{k = 0}^{c-1} \left(\frac{c}{c-k}\right)^{c-k} \cdot \left(\frac{1}{3}\right)^{k} n^{c}\\ &= \sum_{k = 0}^{c-1} \left(1 + \frac{k}{c-k}\right)^{c-k} \cdot \left(\frac{1}{3}\right)^{k} n^{c}\\ &= \sum_{r = 1}^{c} \left(1 + \frac{c-r}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r} n^{c}\\ &= \sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r} n^{c}\\ &= n^{c} \cdot\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}. \end{align}\]

The idea of this part is that we have obtained an expression of the form \(n^c \cdot f(c)\), where \(f(c) = \sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}\). Our goal now is, naturally, to argue that \(f(c)\) is bounded above by a constant. As a first step towards this goal, let us prove first that \(f(c)\) is always increasing. In order to do this consider that

\[\begin{align} f(c+1) - f(c) &= \left(\sum_{r = 1}^{c} \frac{1}{3} \left(\frac{c+1}{c}\right)^{r} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r} - \sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}\right) + 1\\ &= \sum_{r=1}^c \left[\frac{1}{3} \left(\frac{c+1}{c}\right)^{r} - 1\right]\left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r} + 1. \end{align}\]

We now propose the following claim, which is equivalent to \(f\) being increasing.

Claim 1. \(f(c+1) - f(c) > 0.\)

Proof of Claim 1. Because of the previous equation for \(f(c+1) - f(c)\), what we want to prove is equivalent to.

\begin{equation} \sum_{r=1}^c \left[\frac{1}{3} \left(\frac{c+1}{c}\right)^{r} - 1\right]\left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r} > -1. \end{equation} In turn, this is equivalent to proving \begin{equation} \sum_{r=1}^c \left[1 - \frac{1}{3} \left(1+\frac{1}{c}\right)^{r} \right]\left(\frac{3c}{r}\right)^{r} < 3^{c} \end{equation} but as \(\left(1+\frac{1}{c}\right)^{r} > 1\), we have that \begin{equation} \left[1 - \frac{1}{3} \left(1+\frac{1}{c}\right)^{r} \right]\left(\frac{c}{r}\right)^{r} < \left[1 - \frac{1}{3} \right]\left(\frac{c}{r}\right)^{r}, \end{equation} and thus our claim is implied by \begin{equation} \sum_{r=1}^c \left(\frac{3c}{r}\right)^{r} < \frac{3^{c+1}}{2}. \end{equation} Let us prove that this is the case by induction on \(c\). For \(c=1\) it is equivalent to \(3 < \frac{9}{2}\), which obviously holds. For the general case consider that \(\begin{align} \sum_{r=1}^{c+1} \left(\frac{3(c+1)}{r}\right)^{r} &= \left(\sum_{r=1}^{c} \left(\frac{3(c+1)}{r}\right)^{r}\right) + 3^{c+1}\\ &= \left(\sum_{r=1}^{c} \left(\frac{3c}{r}\right)^{r} \left(\frac{c+1}{c}\right)^r\right) + 3^{c+1}\\ &\leq \left(\sum_{r=1}^{c} \left(\frac{3c}{r}\right)^{r}e \right) + 3^{c+1}\\ % \tag{Using that $(1+1/c)^r \leq (1+1/c)^c \leq e$}\\ &\leq e \cdot \frac{3^{c+1}}{2} + 3^{c+1}\\ %\tag{Inductive hypothesis}\\ &= 3^{c+1} \left(\frac{e}{2} + 1 \right)\\ &\leq 3^{c+2}. \end{align}\)

Now let us study the ratio \(f(c+1)/f(c)\). We have that \(\begin{align} \frac{f(c+1)}{f(c)} &= \frac{\sum_{r = 1}^{c+1} \left(\frac{c+1}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c+1-r}}{\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}\\ &= \frac{1}{3} \cdot \frac{\sum_{r = 1}^{c+1} \left(\frac{c+1}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}{\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}\\ &= \frac{1}{3} \cdot \frac{\left(\sum_{r = 1}^{c} \left(\frac{c+1}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}\right) + 3}{\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}\\ &= \frac{1}{3} \cdot \frac{\left(\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \left(\frac{c+1}{c}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}\right) + 3}{\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}. \end{align}\)

Now, using that \((1+\frac{1}{c})^r \leq (1+\frac{1}{c})^c \leq e\) for every positive integer \(c\), we have that \(\begin{align} \frac{f(c+1)}{f(c)} &\leq \frac{1}{3} \cdot \frac{e\left(\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}\right) + 3}{\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}\\ &\leq \frac{e}{3} + \frac{1}{\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r}}. \end{align}\)

Now assume, expecting a contradiction, that there is some positive integer \(C\) such that \(f(C) > 11\). This would imply that \begin{equation} \frac{f(C+1)}{f(C)} \leq \frac{e}{3} + \frac{1}{11} < 0.907 + 0.091 = 0.998 < 1, \end{equation} meaning that \(f(C+1) < f(C)\), which contradicts Claim 1. Therefore, \(f(c) \leq 11\) for every integer \(c\), and thus we have our desired inequality: \begin{equation} \sum_{r = 1}^c \frac{c^r r^c}{r^{2r}} n^r \leq n^{c} \cdot\sum_{r = 1}^{c} \left(\frac{c}{r}\right)^{r} \cdot \left(\frac{1}{3}\right)^{c-r} = n^c \cdot f(c) \leq 11 n^c. \end{equation}