Multinomial sum with positive coefficients

Question

Consider the following sum. $$ \sum_{ \substack{L_1 L_2 ... L_k :\\ L_1 +... + L_k = N\\ \forall i \, \, \, L_i > 0 }}\binom{N}{L_1 , L_2 , ... L_k} $$ It is well known from the Multinomial Theorem that the sum would equal $k^N $ if the restriction $L_i>0$ was replaced by $L_i \geq 0$. Is there a closed expression also for the formula above?

Answer 1

We get from first principles the closed form

$$N! [z^N] \left(\frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots\right)^k.$$

This is

$$N! [z^N] (\exp(z)-1)^k = k! \times N! [z^N] \frac{(\exp(z)-1)^k}{k!} \\ = k! \times {N\brace k}.$$