Hi again, There were some bugs in my implementations. I updated the gist <https://gist.github.com/pabloferz/01675f1bf4c8be359767#file-levicivita-jl> with the corrected versions and added a simpler looking function (but of O(n²) running time).
I did some tests and found (with my slow processor) that for permutations of length <= 5 the quadratic implementation (levicivita_simple) performs as fast as the (levicivita_inplace_check). For lengths from 5 to 15, levicivita_inplace_check is the fastest, followed by levicivita_simple. For lengths from 15 to 25 levicivita_simple and levicivita perform the same (but slower than levicivita_inplace_check). For more than 25 elements levicivita_inplace_check is always the fastest, 2x faster than levicivita and n times faster than levicivita_simple. For people wanting the 3D Levi-Civita tensor, levicivita_simple and levicivita_inplace_check should be the same. For people wanting the parity of a permutation for long permutations levicivita_inplace_check should work the best. Greetings!
