Let $p$ is a positive integer, and let $A_1,\ldots,A_p$ be a partition of $\{1,\ldots,n\}$. My conjecture is that \begin{equation} \displaystyle\sum_{i=1}^p\left( \frac{\displaystyle\sum_{a_i,a_j \in A_p}|a_i-a_j|}{|A_p|} \right)\leq \frac{\displaystyle\sum_{1\leq i,j \leq n}|i-j|}{n}=\frac{(n-1)(n+1)}{3} \tag{1} \end{equation} I have checked that this is true for $n=4,5,6$. I have tried to prove this conjecture by induction, but I did not succeed yet. I think also about the weighted network. The sum $\displaystyle\sum_{a_i,a_j \in A_p}|a_i-a_j|$ is the sum of the inverse of closeness of all points in network represented by $A_p$. But once again, I did not find any good references.
Another way I have thought about the proof is as follows.
Let $A,B$ be two disjoint sets of positive integers. The conjecture above is true if we can prove that \begin{equation} \frac{\displaystyle\sum_{a_i,a_j \in A}|a_i-a_j|}{|A|}+\frac{\displaystyle\sum_{b_i,b_j \in B}|b_i-b_j|}{|B|}\leq \frac{\displaystyle\sum_{x_i,x_j \in A\cup B}|x_i-x_j|}{|A|+|B|} \tag{2} \end{equation} I am trying to prove inequality (2) by proving that if we move one element from set $B$ to set $A$, then it will increase the sum in LHS of (2) provided that $|A|\geq |B|$.
I am still working on it. If you have any ideas, comments, or even counter examples, please do not hesitate to write them here. Thank you very much.
Update: The inequality in (2) seems wrong. I have done a simulation for the case $n=10$. The values of those partitions are not monotone; however, they're all smaller than $\frac{(n-1)(n+1)}{3}$.
