Identifier
-
Mp00064:
Permutations
—reverse⟶
Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001557: Permutations ⟶ ℤ
Values
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,3,2] => [2,3,1] => [3,1,2] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,1,3] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,4,3] => [3,4,2,1] => [4,1,3,2] => [2,4,3,1] => 2
[1,3,2,4] => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,3,4,2] => [2,4,3,1] => [3,1,4,2] => [2,4,1,3] => 2
[1,4,2,3] => [3,2,4,1] => [4,3,1,2] => [3,4,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 2
[2,1,3,4] => [4,3,1,2] => [4,2,3,1] => [4,2,3,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [4,1,3,2] => 0
[2,3,4,1] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 1
[2,4,3,1] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 0
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,2,4] => [2,3,1,4] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [2,1,3,4] => [2,1,3,4] => 0
[3,4,1,2] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 1
[3,4,2,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[4,1,2,3] => [3,2,1,4] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,3,2] => [2,3,1,4] => [1,3,4,2] => [1,4,2,3] => 2
[4,2,1,3] => [3,1,2,4] => [1,4,2,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,1,2] => [2,1,3,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 3
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,4,3,2] => [2,5,4,3,1] => 3
[1,2,4,3,5] => [5,3,4,2,1] => [4,5,3,2,1] => [5,4,3,1,2] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [4,1,5,3,2] => [2,5,4,1,3] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,1,3,2] => [3,5,4,2,1] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [4,5,1,3,2] => [3,5,4,1,2] => 3
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [5,4,2,3,1] => 3
[1,3,2,5,4] => [4,5,2,3,1] => [5,1,3,4,2] => [2,5,3,4,1] => 3
[1,3,4,2,5] => [5,2,4,3,1] => [3,5,4,2,1] => [5,4,1,3,2] => 3
[1,3,4,5,2] => [2,5,4,3,1] => [3,1,5,4,2] => [2,5,1,4,3] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [5,3,1,4,2] => [3,5,2,4,1] => 3
[1,3,5,4,2] => [2,4,5,3,1] => [3,5,1,4,2] => [3,5,1,4,2] => 3
[1,4,2,3,5] => [5,3,2,4,1] => [4,3,5,2,1] => [5,4,2,1,3] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [4,1,3,5,2] => [2,5,3,1,4] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [3,4,5,2,1] => [5,4,1,2,3] => 3
[1,4,3,5,2] => [2,5,3,4,1] => [3,1,4,5,2] => [2,5,1,3,4] => 3
[1,4,5,2,3] => [3,2,5,4,1] => [4,3,1,5,2] => [3,5,2,1,4] => 3
[1,4,5,3,2] => [2,3,5,4,1] => [3,4,1,5,2] => [3,5,1,2,4] => 3
[1,5,2,3,4] => [4,3,2,5,1] => [5,4,3,1,2] => [4,5,3,2,1] => 3
[1,5,2,4,3] => [3,4,2,5,1] => [4,5,3,1,2] => [4,5,3,1,2] => 3
[1,5,3,2,4] => [4,2,3,5,1] => [5,3,4,1,2] => [4,5,2,3,1] => 3
[1,5,3,4,2] => [2,4,3,5,1] => [3,5,4,1,2] => [4,5,1,3,2] => 3
[1,5,4,2,3] => [3,2,4,5,1] => [4,3,5,1,2] => [4,5,2,1,3] => 3
[1,5,4,3,2] => [2,3,4,5,1] => [3,4,5,1,2] => [4,5,1,2,3] => 3
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,2,3,1] => [5,3,4,2,1] => 2
[2,1,3,5,4] => [4,5,3,1,2] => [5,1,4,2,3] => [2,4,5,3,1] => 2
[2,1,4,3,5] => [5,3,4,1,2] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[2,1,4,5,3] => [3,5,4,1,2] => [4,1,5,2,3] => [2,4,5,1,3] => 2
[2,1,5,3,4] => [4,3,5,1,2] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[2,1,5,4,3] => [3,4,5,1,2] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[2,3,1,4,5] => [5,4,1,3,2] => [5,2,4,3,1] => [5,2,4,3,1] => 1
[2,3,1,5,4] => [4,5,1,3,2] => [5,1,2,4,3] => [2,3,5,4,1] => 1
[2,3,4,1,5] => [5,1,4,3,2] => [2,5,4,3,1] => [5,1,4,3,2] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [2,1,5,4,3] => [2,1,5,4,3] => 0
[2,3,5,1,4] => [4,1,5,3,2] => [5,2,1,4,3] => [3,2,5,4,1] => 1
[2,3,5,4,1] => [1,4,5,3,2] => [2,5,1,4,3] => [3,1,5,4,2] => 0
[2,4,1,3,5] => [5,3,1,4,2] => [4,2,5,3,1] => [5,2,4,1,3] => 1
[2,4,1,5,3] => [3,5,1,4,2] => [4,1,2,5,3] => [2,3,5,1,4] => 1
[2,4,3,1,5] => [5,1,3,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => 0
[2,4,3,5,1] => [1,5,3,4,2] => [2,1,4,5,3] => [2,1,5,3,4] => 0
[2,4,5,1,3] => [3,1,5,4,2] => [4,2,1,5,3] => [3,2,5,1,4] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [2,4,1,5,3] => [3,1,5,2,4] => 0
[2,5,1,3,4] => [4,3,1,5,2] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[2,5,1,4,3] => [3,4,1,5,2] => [4,5,2,1,3] => [4,3,5,1,2] => 2
[2,5,3,1,4] => [4,1,3,5,2] => [5,2,4,1,3] => [4,2,5,3,1] => 1
[2,5,3,4,1] => [1,4,3,5,2] => [2,5,4,1,3] => [4,1,5,3,2] => 0
[2,5,4,1,3] => [3,1,4,5,2] => [4,2,5,1,3] => [4,2,5,1,3] => 1
[2,5,4,3,1] => [1,3,4,5,2] => [2,4,5,1,3] => [4,1,5,2,3] => 0
[3,1,2,4,5] => [5,4,2,1,3] => [5,3,2,4,1] => [5,3,2,4,1] => 2
[3,1,2,5,4] => [4,5,2,1,3] => [5,1,3,2,4] => [2,4,3,5,1] => 2
[3,1,4,2,5] => [5,2,4,1,3] => [3,5,2,4,1] => [5,3,1,4,2] => 2
[3,1,4,5,2] => [2,5,4,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => 2
[3,1,5,2,4] => [4,2,5,1,3] => [5,3,1,2,4] => [3,4,2,5,1] => 2
[3,1,5,4,2] => [2,4,5,1,3] => [3,5,1,2,4] => [3,4,1,5,2] => 2
[3,2,1,4,5] => [5,4,1,2,3] => [5,2,3,4,1] => [5,2,3,4,1] => 1
[3,2,1,5,4] => [4,5,1,2,3] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[3,2,4,1,5] => [5,1,4,2,3] => [2,5,3,4,1] => [5,1,3,4,2] => 0
[3,2,4,5,1] => [1,5,4,2,3] => [2,1,5,3,4] => [2,1,4,5,3] => 0
[3,2,5,1,4] => [4,1,5,2,3] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [2,5,1,3,4] => [3,1,4,5,2] => 0
[3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => [5,2,1,4,3] => 1
[3,4,1,5,2] => [2,5,1,4,3] => [3,1,2,5,4] => [2,3,1,5,4] => 1
[3,4,2,1,5] => [5,1,2,4,3] => [2,3,5,4,1] => [5,1,2,4,3] => 0
[3,4,2,5,1] => [1,5,2,4,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[3,4,5,1,2] => [2,1,5,4,3] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[3,4,5,2,1] => [1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 0
[3,5,1,2,4] => [4,2,1,5,3] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[3,5,1,4,2] => [2,4,1,5,3] => [3,5,2,1,4] => [4,3,1,5,2] => 2
[3,5,2,1,4] => [4,1,2,5,3] => [5,2,3,1,4] => [4,2,3,5,1] => 1
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Tanimoto
Description
Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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