What is the probability that $\pi(x) + x$ is injective?

Solution 1:

My laptop has extended your results to $n=3$ and $n=4$ (but $n=5$ is completely out of reach).

For $n=3$ we have $8!=40320$ permutations, of which $384$ induce an injection. $384/40320 = 1/105$.

For $n=4$ we have $16!=20922789888000$ permutations, of which $244744192$ induce an injection. $244744192/20922789888000 = 97/8292375$ in lowest terms, so I don't see a nice pattern here.

Solution 2:

There is a name for what you describe: it's called an orthomorphism. If $S$ is the cyclic group of order $n$, then the number of orthomorphisms for $n=1,3,5,...$ is $1, 3, 15, 133, 2025, 37851,...$ (sequence A006717 in OEIS). (If $n$ is even, $S$ has no orthomorphisms.)

You can find lots of links for the case when $S$ is GF$(2^n)$ by googling "orthomorphism of galois field". For instance, here is a short paper which confirms my $244744192$ and goes on to discuss the construction of permutation polynomials.