What is the closed-form expression for the summation
$$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$, $m$ and $p$? Here $\alpha=(\alpha_1,\cdots,\alpha_n)$ is a $n$-tuple of non-negative integers.
This equation can also be seen as multinomial expansion without coefficients.
The answer to the question asked by xivaxy: Multi-index power series doesn't work when $p_i$ = $p_j$ and $i$ != $j$.