Let N denote the positive integers. For any permutation p: N ® N, let T(p): N ® N be given by T(p)(n) = # of elements in {m Î N | n £ m and p(m) £ p(n)}. Observe that T is a bijection from the set of permutations N ® N onto the set of sequences N ® N that contain infinitely many 1s.
Now suppose f: N ® N contains infinitely many 1s; then its Quet transform Q(f): N ® N is T ([T -1(f)]-1) , which also contains infinitely many 1s. Q is self-inverse; f and Q(f) correspond via T to a permutation and its inverse.
Leroy Quet's example: let f be A002260:
1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5...
then T -1(f) is A065562:
1, 2, 4, 3, 6, 8, 5, 9, 11, 13, 7, 12, 15, 17, 19...
The inverse of this permutation is A065579:
1, 2, 4, 3, 7, 5, 11, 6, 8, 16, 9, 12, 10, 22, 13...
so Q(f) is A101387:
1, 1, 2, 1, 3, 1, 5, 1, 1, 7, 1, 2, 1, 9, 1...