I'm considering a discrete problem with integer vectors, and I don't know whether there exists some literature about such problems. Perhaps somebody recognizes the structure of this problem.
So, suppose I am considering some integer $n$ and correspondingly the vector $v = [1,\ldots,n]^T$. If I now take any permutation matrix $P$, I also consider the permuted vector $Pv$. Now I am considering the difference: $$v - Pv$$ In particular, I am interested in either the $1$-norm or $2$-norm of this vector, so: $$||v-Pv||_1 \quad \text{or} \quad ||v-Pv||_2.$$ What I want to know is the average of either (any of both would be fine) of this quantities of this difference vector over all possible permutations $P$, as an expression in $n$.
If possible, I would even like to know something about the distribution of this quantity, i.e. the likelihood of of finding some value for $||v-Pv||_1$ in case for the $1$-norm. But that would be an additional question.