I want to rank and unrank permutations with one cycle in lexicographical order with a given len.
A permutation with one cycles is where you can visit in this cycle each element.
p:= (2,3,1) is a permutation with one cycle. Has rank 1.
p:= (3,1,2) has 1 cycle too, but rank 2, because the permutation is lexicographical greater the frist so it becomes a greater rank.
p:= (1,2,3) is a permutation with 3 cycles. (1),(2),(3)
How can I efficently rank (permutation with one cycle to rank) and unrank (rank + len to permutation with one cycle) in lexicographical order? I have no idea how to archive this.
I discovered a solution for ranking. We know that a permutations of length n has n-1! permutations with one cycle. Due to this knowledge we can come to the following solution.
Ranking: example 2341
We start to calculating the rank with the 1 position this gives (n-1[position])! as tempvalue. Then we calculating the index of the 2 which is 0, because 1 is falling out through it creates the cycle (1). To complete the calculating for the first position we need to multiply the index of the element with the tempvalue, which leads to 0 as temprank_0.
Now we continue this steps for the remaining positions to add temprank_0+temprank_1+temprank_2+temprank_4 = 0
Unranking: 4 for permutation len 4:
We divide the rank through (n-2[postion+1])! which leads 2 which is the index of 1234 which dont create a cycle so the 1 position of the permutation is 4 . Then we subtract 2 times (n-2)!from 4.
This we continue this twice. So we have 412. So in the end we add just the remaining value 3 end we get the permutation 4123 .