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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.

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  • $\begingroup$ Draw a picture (this is almost always a good idea with problems like these). Draw vertical lines for the sums in the original partition, and horizontal ones for the conjugate position. The cells where the lines cross are in both sums, so all you need to do is to look at the fringe. $\endgroup$ – deinst Apr 8 '13 at 13:14
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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:

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

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