Mean Value of Permutation Sums

Problem Let \((a_{1},a_{2}, \cdots ,a_{n})\) be a permutation of the set \(\{1,2, \cdots,n \}\). Compute the average value, indicated with \(M_{n}\) , of the following sum: \[ (a_{1}- a_{2})^2 + (a_{2}-a_{3})^2 + \cdots + (a_{n-1} – a_{n})^2 \] taken on all permutations. If \(n=2\) the set of permutations is \(\{12,21\}\) Read more…