What are the examples of the easily computable "wild" permutations?
Yes, such things exist.  They have been studied in the cryptographic literature: the key name is "format-preserving encryption".

To learn about this subject, I suggest taking a look at https://en.wikipedia.org/wiki/Format-preserving_encryption and https://crypto.stackexchange.com/q/16561/351 and https://crypto.stackexchange.com/q/504/351 and https://crypto.stackexchange.com/q/20035/351 and https://crypto.stackexchange.com/q/18988/351 and 
https://crypto.stackexchange.com/questions/tagged/format-preserving.  You could also construct a suitable block cipher yourself using a Feistel construction, but it's probably better to use a standard scheme that has been analyzed in the literature.

There are many constructions.  The best one will depend on the specific value of $n$ in your application.  Take a look at the schemes in the cryptographic literature, then if you have a more specific question about how to use one of those schemes, you can ask either here or on Crypto.SE.