When does the following hold?

$\sum_{(i_1,\ldots,i_k)\in E} \frac{n!}{i_1! \ldots i_k!} \le \exp(n H^*)$

Where

$H^*=\max_{(i_1,\ldots,i_k)\in E} -(\frac{i_1}{n}\log \frac{i_1}{n}+\ldots +\frac{i_k}{n}\log \frac{i_k}{n})$

Motivation: this is a generalization of Chernoff's bound to fair k-sided dice