Ratio of collision of rate between distributions?

Background

Consider the familiar setting of statistical mechanics, an assembly of $N=4$ distinguishable gas particles. Let, the total energy be $4E_0$.

The possible ways energy of sharing energy between the particles gives us $5$ possible distributions $D(i)$:

$$begin{vmatrix}
& 0 &E_0 & 2E_0 & 3E_0 & 4E_0
D(1) & 3 & 0 & 0 &0 & 1
D(2) & 2 & 1 & 0 & 1 & 0
D(3) & 2 & 0 & 2 & 0 & 0
D(4) & 1 & 2 & 1 &0 &0
D(5) & 0 & 4 & 0 &0 & 0
end{vmatrix}
$$

The number of microstates in $D(i)$ is given by:

$$ t(D(i)) = frac{N!}{prod n_k!}$$

There are $35$ micro-states in total in this example. As,

$$ t(D(1))+ t(D(2)) + t(D(3)) + t(D(4)) + t(D(5)) = 4 +12 + 6+ 12 + 1 =35$$

Since, each microstate is equally probable (at equilibrium) we can take time unit $ 35 Delta t$ then we can state on average the distributions are stable for time:

$$begin{vmatrix}
D(1) & 4 Delta t
D(2) & 12 Delta t
D(3) & 6 Delta t
D(4) & 12 Delta t
D(5) & Delta t
end{vmatrix}
$$

Now, let us only keep track of collisions that change the (energy) distribution. For example, imagine an $D(1) to D(2)$ where $4E_0 + 0 to E_0 + E_3$. We restrict ourselves $2$ particles per collision. Then following transitions (between distributions) are only are only allowed.

$$ D(1) to D(2) , D(3)$$
$$ D(2) to D(3) , D(1) , D(4)$$
$$ D(3) to D(1) , D(2) , D(4)$$
$$ D(4) to D(2) , D(3), D(5) $$
$$ D(5) to D(4) $$

Question

Is there nice way to calculate the ratio of collision rate that go from $D(i) to D(j):D(k) to D(l) $? Is there generalizable version for an arbitrary number particles?