Question: I am trying to draw the Young diagram for $S\_5$. I know the following pieces of information about $S\_5$.

1. The order of the group is $120$.
2. The number of conjugacy classes and so partitions is $7$.
3. Degrees of irreducible representations $1,1,4,4,5,5,6$.
4. The partition is $1 + 10 + 15 + 20 + 20 + 24 + 30 = 120$.

I understand that the Young diagram should contain $30$ boxes in the first row, $24$ boxes in the second row, $20$ boxes in the third and fourth rows, $15$ boxes in the fifth row, $10$ boxes in the sixth row and $1$ box in the seventh row.

So, the Young diagram is as follows.

My question:

Am I doing it right? I understand that the next step is to fill up the boxes to make it a Young tableau.

[![enter image description here](https://i.stack.imgur.com/aTz5v.png)](https://i.stack.imgur.com/aTz5v.png)

**UPDATE 1:**

I was able to compute the partition as follows.

[![enter image description here](https://i.stack.imgur.com/S6zwl.png)](https://i.stack.imgur.com/S6zwl.png)

What should be my next step?

**UPDATE 2:**

I think I was able to draw the Young diagrams.

[![enter image description here](https://i.stack.imgur.com/6bGQw.png)](https://i.stack.imgur.com/6bGQw.png)

Answer: 
$$
\int\sin(\pi t)\cos(\pi t)\,dt=
\frac{1}{2}\int\sin(2\pi t)\,dt=-\frac{\cos(2\pi t)}{4\pi}+c
$$
Just check the derivative.

Now you can expand $\cos(2\pi t)$ as
$$
\cos(2\pi t)=2\cos^2(\pi t)-1=1-2\sin^2(\pi t)
$$
and get both forms. None is “the right one”, both are correct. Functions which differ by a constant have the same derivative.
Notice, both the results obtained are really same just need to manipulate as follows $$\int \sin(\pi t)\cos(\pi t)\ dt$$ $$=\int \frac{1}{2}(2\sin(\pi t)\cos(\pi t))\ dt$$
$$=\frac{1}{2}\int\sin(2\pi t)\ dt $$
$$=\frac{1}{2}\left(-\frac{1}{2\pi}\cos (2\pi t)\right)+c$$
$$=-\frac{2\cos^2(\pi t)-1}{4\pi}+c$$
$$=-\frac{2\cos^2(\pi t)}{4\pi}+\frac{1}{4\pi}+c$$
substituting a new constant $\frac{1}{4\pi}+c=C$, 
$$=-\frac{\cos^2(\pi t)}{2\pi}+C$$