# 1 is to q as n! is to…

If one believes that $[n]=\{1,\dots,n\}$ is a vector space over the field with one element $\mathbb{F}_1$, then a permutation of $[n]$ is just a complete flag. Concretely, a permutation $(x_{\sigma(1)},\dots,x_{\sigma(n)})$ corresponds to the chain of subspaces $\{x_{\sigma(1)}\}\subseteq \{x_{\sigma(1)}, x_{\sigma(2)}\}\subseteq\dots\subseteq \{x_{\sigma_1}, x_{\sigma(2)},\dots,x_{\sigma(n)}\}$.

As we all know, there are $n!$ permutations of $[n]$, or equivalently complete flags of $\mathbb{F}_1^n$. Give a formula for the number of complete flags of $\mathbb{F}_q^n$, which will be a $q$-analogue for the factorial.

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?