Combinatorial interpretation of a sum with Stirling numbers

Question

$$\sum_{i,j}{n\brack i+j}\binom{i+j}i$$ Does this have a combinatorial interpretation? I don't see how to use Stirling numbers of the first kind in interpretations. I know that the answer is $(n+1)!$ , but the original question didn't provide it.

Answer 1

I'm not sure about a direct combinatorial proof, but the solution can be easily obtained by taking $k=i+j$ and using the known formula $\sum_{k} \genfrac[]{0pt}{}{n}{k} x^k = x^{\overline{n}}$, where $x^{\overline{n}}$ denotes the rising factorial.

$$\sum_{i,j} \genfrac[]{0pt}{}{n}{i+j} \binom{i+j}{i} = \sum_{k} \genfrac[]{0pt}{}{n}{k} \sum_{i} \binom{k}{i} = \sum_{k} \genfrac[]{0pt}{}{n}{k} 2^k = 2^{\overline{n}} = (n+1)!$$