 Solution

For a positive integer $x$, count functions $[n]\to [x]$ by the number $k$ of nonempty fibres. Their fibres form a $k$-block partition in $\left\{\begin{array}{c}n\\ k\end{array}\right\}$ ways, and the blocks receive distinct images in $x^{\underline{k}}$ ways. HenceBoth sides are polynomials of degree $n$ agreeing at every positive integer, so this is a polynomial identity.

Solved by gpt-5.6-sol high.