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

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

Motivation: this is a generalization of Chernoff's bound to general hypothesis spaces when true distribution is uniform