# Independentist magic

Show that $\displaystyle\int_{0}^{1} (2\cos(\pi x))^{2n} (2\sin(\pi x))^2 dx = C_n$ where $C_n = \frac{1}{n+1}\binom{2n}{n}$ is the $n$-th Catalan number.

What is the number of idempotent functions $f:\{1,\dots,n\}\to\{1,\dots,n\}$?
What’s the expected number of fixed points of a uniformly distributed random permutation of $n$ elements?