Why does this formula for the partition function not include the multiplicity? I am having problems understanding the formulas used for describing the partition functions and the probability distributions for canonical ensembles.
In the first case I have two formulas for the partition function:
I can label each system microstate with $j$, associate it with an energy $E_j$, and state that:
$$p_j = e^{-\beta E_j/Z}\quad\text{with}\quad Z=\sum_je^{-\beta E_j/Z}$$
On the other hand, in my lecture notes, the canonical partition function and the probability distribution are given by:
$$\begin{align}
Z &= \sum_j\Omega(N,Q)e^{-\beta E_j/Z} \\
P(Q) &=(1/Z)\Omega(N,Q)e^{-\beta Q}
\end{align}$$
for a given system of $N$ weakly coupled oscillators and $Q$ quantas.
My question is: Why is the multiplicity $\Omega(N,Q)$ not taken into account in the first case?
 A: (I am not sure this is an answer but it is to long to be a comment)
Let us create a simple example of a system of $3$ states, state $1$, state $2$ and state $3$. Let state $1$ and state $2$ both have an energy of $E$ and state $3$ have an energy of $E'\ne E$.
Your first summation is summing over individual states. I.e. it is saying 'let us call the energy of state $1$; $E_1$,  the energy of state $2$; $E_2$ and the energy of state $3$; $E_3$. The sum then looks something like this:
$$Z=\sum_{j=1}^3e^{-\beta E_j}$$
$$=e^{-\beta E_1}+e^{-\beta E_2}+e^{-\beta E_3}$$
$$=2e^{-\beta E}+e^{-\beta E'}$$
Whilst your second summation is summing over individual energies. I.e. it is saying 'let us call the energy $E$; $E_1$ and the energy $E'$; $E_2$. With $\Omega_1=2$ and $\Omega_2=1$ (i.e. the number of states with each energy). our sum now looks something like this:
$$Z=\sum_{j=1}^2 \Omega_j e^{-\beta E_j}$$
$$=2e^{-\beta E}+e^{-\beta E'}$$
I hope this clears it up a bit, let me know if you have any further problems. 
