Given a multinomial distribution with parameters $n>0$ where $n$ is an integer and event probabilities $p_i= 1/k$ for $i \in \left\{1, \ldots, k\right\}$.
Next, allow that $\mathbf{N}$ is a diagonal matrix in $\mathbb{R}^{k\times k}$ such that diagonal entries of $\mathbf{N}$ are comprised of the support of the multinomial distribution. So, the $i$th diagonal element of $\mathbf{N}$ is $n_{i,i} \in \left\{0, \ldots, n\right\}$ for $i \in 1,\ldots, k$; and $\sum\limits_{j=1}^k{n_{i,i}} = n$.
My Questions:
1) In closed form, what is the expected value of the determinant of $\mathbf{N}$, $E{(\det{(\mathbf{N})})}$ for $n>0$ and any $k \in \left\{1,\ldots, n \right\}?$ And 2) If a closed-form expression is not forthcoming, what is an upper and lower bounding for $E{(\det{(\mathbf{N})})}$ for $n>0$ and any $k \in \left\{1,\ldots, n \right\}?$
Example 1
For the special case of $k=2$, this problem reduces to finding the expected value $E{(X_1, X_2)}$, which is related to the covariance. Given that stated parameters that $p_1 = p_2$, I have derived $$E{(\det{(\mathbf{N})})} = \dfrac{n\,(n-1)}{2^2}.$$
Example 2
For the special case of $k=n$, this problem is also soluble. Because of all the $p_n{(n)}$ partitions of $n$ into $n$ parts, there is only one partition that leads to a non-zero product. Given that stated parameters, I have derived $$E{(\det{(\mathbf{N})})} = \dfrac{n!}{n^n}.$$