Using Wick's Theorem in an example with the harmonic oscillator I understand Wick's theorem to be,
$$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$
And I'm researching combinatorics and quantum theory in general.
How would one connect Wicks theorem to the quantum HO, an example would be appreciated.
 A: I'm not sure what you mean by this notation.  I think most readers will be more comfortable with
$$T\{\phi(x_1)\phi(x_2)...\phi(x_m)\}=N\{\phi(x_1)\phi(x_2)...\phi(x_m) + \text{all possible contractions}\}.$$
For general operators $\hat{A}, \hat{B}, \hat{C},...,\hat{Z}$, you could write
$$T\{\hat{A}\hat{B}\hat{C}...\hat{Z}\}=N\{\hat{A}\hat{B}\hat{C}...\hat{Z} \text{ }+ \text{all possible contractions of} \text{ } \hat{A}\hat{B}\hat{C}...\hat{Z}\}.$$
The notation $N\{\}$ stands for normal ordering and it is the same as ": :" - you may use it if you like it.
I have never heard of Wick's theorem in relation with the quantum harmonic oscillator.  Usually, Wick's theorem is used for evaluating $S$-matrix elements, in which vacuum expectation values of time-ordered strings of operators appear:
$$\langle 0|T\{\hat{A}\hat{B}\hat{C}...\hat{Z}\}|0 \rangle.$$
Using Wick's theorem, it becomes much easier to calculate such objects.  I will let you find out why!
A: 
I understand Wicks theorem to be,


$$T(x)=\mathcal{N}(x)=\sum:\textbf{all contractions}:$$

No, the above is incorrect. The time ordering and normal ordering are not generally the same, which is why Wick's theorem is useful.

And I'm researching combinatorics and quantum theory in general.
How would one connect Wicks theorem to the quantum HO, an example would be appreciated.

In studying the quantum harmonic oscillator, we introduce lowering (annihilation) and raising (creation) operators:
$$
\hat a
$$
and
$$
\hat a^\dagger\;,
$$
respectively. We have, as usual, $\hat a|0> = 0$ and $[a,a^\dagger] = 1$ and $\hat H = \hat a \hat a^\dagger (\hbar \omega_0 + 1/2)$.
These $a$ operators have time dependence in the Heisenberg picture (in the same way as all operators):
$$
\hat a(t) = e^{iHt}\hat a e^{-iHt}\;,
$$
etc.
The time ordering, for example, of $a(t)$ and $a^\dagger(t')$ is:
$$
T(a(t)a^\dagger(t')) = \theta(t-t')a(t)a^\dagger(t') + \theta(t'-t)a^\dagger(t')a(t)\;.
$$
The normal ordering, for example, of $a(t)$ and $a^\dagger(t')$ is:
$$
N(a(t)a^\dagger(t')) = a^\dagger(t')a(t)\;.
$$
Wick's theorem can be used to relate these.
For example:
$$
<0|T(a(t)a^\dagger(t'))|0> = \theta(t-t')e^{-i\hbar \omega_0 (t-t')} + <0|N(a(t)a^\dagger(t'))|0>
$$
