# Permutations with the same sum

Note: Rearrangement Inequality is

$$x_ny_1 + \ldots + x_1y_n \leq x_{\sigma(1)}y_1+ \ldots + x_{\sigma(n)}y_n \leq x_1y_1+\ldots + x_ny_n$$

for every choice of real numbers

$$x_1 \leq \ldots \leq x_n$$ and

$$y_1 \leq \ldots \leq y_n$$

and every permutation $x_{\sigma(1)}, \ldots, x_{\sigma(n)}$ of $x_1, \ldots , x_n$.

I need to find all permutations which gives the same sum.

For example, for $$\sigma =\begin{pmatrix} 1& 2& 3& 4& 5& 6\\ 2& 5& 6& 3& 4& 1 \end{pmatrix}, \omega =\begin{pmatrix} 1& 2& 3& 4& 5& 6\\ 6& 3& 4& 1& 2& 5 \end{pmatrix}, \theta =\begin{pmatrix} 1& 2& 3& 4& 5& 6\\ 6& 1& 4& 5& 2& 3 \end{pmatrix}$$ The sum is $\sum_{i=1}^{6} i\cdot \sigma(i)= \sum_{i=1}^{6} i\cdot \theta(i)= \sum_{i=1}^{6} i\cdot \omega(i)=68$.

Question: Find the condition for permutations $\sigma, \theta$ for which $\sum_{i=1}^{n} i\cdot \sigma(i)= \sum_{i=1}^{n} i\cdot \theta(i)$.

• I am no expert of this topic, but this seems related to the inversion statistics of permutations, see this, this, this and this. – Sangchul Lee Feb 28 '18 at 6:26