Number of combinations to make a sequence of length $N$ using set of $K \leq N$ elements so that each element occurs at least one time.
For example with set ${1, 2}$ and $N = 3$ all possibilities are:
- (1, 1, 2)
- (1, 2, 1)
- (1, 2, 2)
- (2, 1, 1)
- (2, 1, 2)
- (2, 2, 1)
What equals to: $\binom32\binom11 + \binom11\binom32$
Is there any way to get expression for number of combinations through $N$ and $K$?
Starting with the combinatorial class
$$\def\textsc#1{\dosc#1\csod} \def\dosc#1#2\csod{{\rm #1{\small #2}}} \textsc{SEQ}_{=K}(\textsc{SET}_{\ge 1}(\mathcal{Z}))$$
we get the generating function
$$G(z) = (\exp(z)-1)^K$$
with the closed form using Stirling numbers
$$N! [z^N] (\exp(z)-1)^K = N! [z^N] K! \frac{(\exp(z)-1)^K}{K!} = K! {N\brace K}.$$