Twelvefold way

Summary

The twelvefold way is a table of twelve basic counting problems, see the definition. Each problem counts maps from a finite set $N$ to a finite set $X$. The maps may be arbitrary, injective or surjective. They may be counted as they are, or up to permutations of $N$, of $X$, or of both. The twelve answers bring together powers, falling factorials, binomial coefficients, Stirling numbers of the second kind and integer partitions.