The following algorithm generates the next permutation lexicographically after a given permutation. Sets on bottom of recursion correspond to prefixes in the permutation. The Shannon entropy of such a random permutation is bits. Also as a base for optimal hashing in Unique Permutation Hashing. Now we obtain equivalence of the two definitions of parity of a permutation. Monte Carlo methods are also a compromise between approximate randomization and permutation tests. Another twelve identities follow by cyclic permutation. There are several short proofs of the invariance of this parity of a permutation. Another eight identities follow by cyclic permutation. In classical music, Bach used the symmetry concepts of permutation and invariance. Any cyclic permutation of is associated to one of the above fractions. Notice the cyclic permutation of indices in this equation&colon; formula 31. They tried many approaches including "knapsack-based" and "permutation polynomials". The value of the determinant is the same as the parity of the permutation. The procedure adopted is a kind of randomization or permutation test.