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.
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:
$$\begin{array}{l|l} \begin{array}{ccc} \pi\\ \hline \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet\\ \bullet&\bullet&\bullet\\ \bullet&\color{red}{\bullet}\\ & \end{array}& \begin{array}{ccc} \pi'\\ \hline \bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\ \bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&\color{red}{\bullet}&\bullet&\color{red}{\bullet}\\ \bullet&\bullet&\bullet&\bullet\\ \bullet&\color{red}{\bullet}&\bullet\\ \bullet \end{array} \end{array}$$