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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 proof 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.

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

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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=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))$. Then proceed in the same manner as the the analogous proof for binomial coefficients, substituting this inequality in the last step of derivation. Note that if any$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$'s. Black dot is the highest entropy coefficient. 1 [made Community Wiki] Take arbitrary non-negative reals$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$Sum of multinomial coefficients in this set is bounded by the highest entropy one. Proof, define$q_i=-\log a_i$and observe that${q_1}^{i_1} \cdots {q_n}^{i_n}\ge \exp(-n H(q))$. Then proceed in the same manner as the the analogous proof for binomial coefficients, substituting this inequality in the last step of derivation. Note that if any$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.