My question is related to the sum 


\begin{equation}
S(n,N) = \sum\_{k\_1+k\_2+...+k\_N=n}\frac{n!}{(k\_1!)\cdot(k\_2!)\cdot...\cdot(k\_N!)} = N^n,
\end{equation}


which is comes from the multinomial theorem: <https://en.wikipedia.org/wiki/Multinomial_theorem>.


Recently I came across the very similar expression


\begin{equation}
\tilde{S}(n,N) = \sum\_{\substack{k\_1+k\_2+...+k\_N=n\\k\_1+2\cdot k\_2+...+N\cdot k\_N=N}}\frac{n!}{(k\_1!)\cdot(k\_2!)\cdot...\cdot(k\_N!)},
\end{equation}


which has an extra restriction $\sum\_{i=1}^N i\cdot k\_i=N$ on the summation indices $k\_i$. I'm not sure how to approach this when trying to evaluate $\tilde{S}$, so any input and answers are much appreciated!



Summation of multinominal coefficients with extra bounds on summation indices
This is a sum of coefficients of $t^N\prod x\_i^{k\_i}$ in $(tx\_1+t^2x\_2+\dots)^n$. In other words, it is the coefficient of $t^N$ in $$(t+t^2+\dots)^n=t^n (1-t)^{-n}=\sum \binom{-n}{k}(-1)^k t^{k+n},$$
hence your sum equals $$\binom{-n}{N-n}(-1)^{N-n}=\binom{N-1}{N-n}.$$


