I have the following product of monomials in $x$:
\begin{align*} \prod_{k=1}^n \left( \sum_{1\leq i_1 < \cdots < i_k\leq n} x_{i_1} \cdots x_{i_k} \right)^{m_k}, \end{align*} where $\sum_{k=1}^n k m_k = N$ for some $N\geq n$ und the $m_k$ are supposed to be non-negative integers. This turns out to be a homogeneous polynomial of degree $N$. I was wondering if there is good way to write this as a sum of monomials like \begin{align*} \sum_{l_1 + \cdots + l_n = N} C_{l_1,\dots,l_n} x_1^{l_1} \cdots x_n^{l_n}. \end{align*} I want to write down the coefficients $C_{l_1,\dots,l_n}$ explicitly, so I hope that someone has seen this before and can help me.