Conjugate Ferrers diagrams 
Let $\pi=\langle \pi_1,\pi_2,... \rangle , \ \pi_1\ge\pi_2\ge...,$ be a partition of a number and $\pi'=\langle \pi_1',\pi_2',... \rangle$ be a partition conjugated to $\pi$, which means that Ferrers diagram for $\pi'$ is transposed Ferrers diagram for $\pi$. For example partition conjugated to $\langle 4,4,2,1 \rangle$ is $\langle 4,3,2,2 \rangle$.
Prove identities:

*

*$\displaystyle\sum_{i}\left\lceil \frac{\pi_{2i-1}}{2} \right\rceil = \sum_{i}\left\lceil \frac{\pi'_{2i-1}}{2} \right\rceil$

*$\displaystyle\sum_{i}\left\lfloor \frac{\pi_{2i-1}}{2} \right\rfloor = \sum_{i}\left\lceil \frac{\pi'_{2i}}{2} \right\rceil$

*$\displaystyle\sum_{i}\left\lfloor \frac{\pi_{2i}}{2} \right\rfloor = \sum_{i}\left\lfloor \frac{\pi'_{2i}}{2} \right\rfloor$

No idea how to even start. Nice observation is that $\pi_1'$ is the number of the elements in $\pi$ but it gives us nothing I think.
 A: HINTS: If you take the first, third, fifth, etc. of a set of $n$ elements, you end up taking $\left\lceil\frac{n}2\right\rceil$ elements; if you take the second, fourth, sixth, etc., you end up taking $\left\lfloor\frac{n}2\right\rfloor$ elements
First identity:
$$\begin{array}{l|l}
\begin{array}{ccc}
\pi\\ \hline
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet\\
&
\end{array}&
\begin{array}{ccc}
\pi'\\ \hline
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet\\
\color{red}{\bullet}
\end{array}
\end{array}$$
Second identity:
$$\begin{array}{l|l}
\begin{array}{ccc}
\pi\\ \hline
\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet\\
&
\end{array}&
\begin{array}{ccc}
\pi'\\ \hline
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet&\bullet\\
\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet
\end{array}
\end{array}$$
Third identity:
$$\begin{array}{l|l}
\begin{array}{ccc}
\pi\\ \hline
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\
\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}\\
&
\end{array}&
\begin{array}{ccc}
\pi'\\ \hline
\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\
\bullet&\bullet&\bullet&\bullet\\
\bullet&\color{red}{\bullet}&\bullet\\
\bullet
\end{array}
\end{array}$$
