# It’s counting time

What is the number of idempotent functions $f:\{1,\dots,n\}\to\{1,\dots,n\}$?

$\sum_\sigma (-1)^\sigma t^{\mathrm{Fix}\;\sigma} = \det\begin{pmatrix} t & 1 &\cdots& 1& 1\\ 1 & t &\cdots& 1& 1 \\ \vdots & \vdots &\ddots & \vdots & \vdots \\ 1 & 1 &\cdots& t& 1\\ 1 & 1 &\cdots& 1& t \end{pmatrix}$
the sum running through $S_n$ and the matrix being of size $n\times n$.