I'm wondering if there's a special polynomial with a name out there with $x_1,x_2,\ldots,x_k$ as variables that's defined like this:
$ \sum_{\substack{i_1>0,i_2>0, \ldots,i_k>0 \\ i_1 +i_2+\cdots+i_k=n}} \frac{n!}{i_1! i_2! \cdots i_k!} x_1^{i_1} x_2^{i_2}\cdots x_k^{i_k} $