This question is prefaced by this part:
We use the notation S(k,n) to stand for the number of partitions of a k element set with n blocks. For historical reasons, S(k,n) is called a Stirling Number of the second kind.
In a partition of the set [k], the number k is either in a block by itself, or it is not. How does the number of partitions of [k] with n parts in which k is in a block with other elements of [k] compare to the number of partitions of [k - 1] into n blocks? Find a two-variable recurrence for S(k,n), valid for k and n larger than one.
So if I understand this correctly the answer will be a proof? Can I reach the answer by enumerating different example partitions of sets? I appreciate any tips and help. Thank you