Growth Rate $n\ln n$

Question

I mistakenly posted this on MathOverflow. I hope this is a better place for it.

I have been investigating a problem about sports teams and came across the function $n\ln(n)$. I want to see if I can find a connection between my problem and other mathematics through this function. Where else does this function occur? Are there other common situations?

Someone suggested the Coupon Collector Problem. Are there other situations?

Answer 1

Let $p_n$ be the $n$-th prime. The Prime Number Theorem tells us that $$\lim_{n\to\infty} \frac{p_n}{n\ln n} = 1. $$

Answer 2

Efficient sorting of $n$ random numbers with an unknown distribution is expected to take a time proportional to $n \log n$.

The logarithm of $n!$ is slightly less than $n \log n$