Take arbitrary non-negative reals $a_1,\ldots,a_n$ consider and consider set $E$ of n-tuples $(i_1,\ldots,i_n)$ satisfying the following

$i_1 a_1 + \ldots + i_n a_n \le n(a_1 \exp - a_1 + \ldots +a_n \exp -a_n), \sum_k i_k=n, i_k\ge 0$

Sum of multinomial coefficients in this set is bounded by the highest entropy one.

Proof, define $q_i=-\log a_i$ and observe that ${q_1}^{i_1} \cdots {q_n}^{i_n}\ge \exp(-n H(q))$. Then proceed in the same manner as the the analogous <a href="http://yaroslavvb.com/upload/binomial.png">proof</a> for binomial coefficients, substituting this inequality in the last step of derivation.

Note that if any $a_i$ is 0, the bound is vacuous. Here are some examples of sets of trinomial coefficients defined in this way for random $a_i$'s. Black dot is the highest entropy coefficient.

<img src="http://yaroslavvb.com/upload/multinomial-sets.gif">

One way of to think of these sets is the following -- view multinomial coefficients as multinomial probability distributions. Then I pick my "highest entropy" distribution q and consider the set of distributions $p$ for which $p_1 \log q_1 + \ldots p_n \log q_n \ge q_1 \log q_1 + \ldots + q_n \log q_n$. That includes q, and all distributions "further away" from the uniform distribution than q