A combinatorial identity

Question

I asked this question in math overflow, but I didn't get a satisfied answer. So I ask here again. Given two positive integers $k,n$, let $\{m_{r}(c)\}_{1\leq r\leq k,1\leq c\leq n}$ be a sequence of non-negative integers satisfy $\sum\limits_{1\leq r\leq k\\1\leq c\leq n}rm_{r}(c)=k$. And suppose $\chi(1),\chi(2),\dots,\chi(n)$ be $n$ variables. Then I believe the following identity should hold: \begin{equation} \sum_{\sum\limits_{1\leq r\leq k,1\leq c\leq n}rm_{r}(c)=k}\prod_{1\leq c\leq n}\chi(c)^{\sum\limits_{r}m_{r}(c)}\sharp\{m_{r}(c)\}_{1\leq r\leq k,1\leq c\leq n}=(\sum\limits_{1\leq c\leq n}\chi(c))(\sum\limits_{1\leq c\leq n}\chi(c)+n)\dots(\sum\limits_{1\leq c\leq n}\chi(c)+(k-1)n) \end{equation} where for each sequence $\{m_{r}(c)\}_{1\leq r\leq k,1\leq c\leq n}$ we define $\sharp\{m_{r}(c)\}_{1\leq r\leq k,1\leq c\leq n}=\frac{n^{k}k!}{\prod\limits_{c,r}((nr)^{m_{r}(c)}m_{r}(c)!)}$. But I don't know how to prove it.