Identifier
-
Mp00090:
Permutations
—cycle-as-one-line notation⟶
Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤ
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [2,3,1] => [1,2,3] => 0
[3,2,1] => [1,3,2] => [2,3,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => [1,2,4,3] => 0
[3,2,1,4] => [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 1
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => [1,2,3,4] => 0
[4,2,1,3] => [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 1
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => [1,2,3,4] => 0
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => [1,2,3,4] => 0
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1
[2,5,1,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[2,5,3,1,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[2,5,4,1,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[3,1,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,1,2,5,4] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,1,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [1,2,4,3,5] => 0
[3,1,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => [1,2,5,4,3] => 0
[3,1,5,2,4] => [1,3,5,4,2] => [3,5,2,4,1] => [1,3,2,5,4] => 1
[3,1,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => [1,2,3,4,5] => 0
[3,2,1,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,2,1,5,4] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,2,4,1,5] => [1,3,4,2,5] => [2,4,1,3,5] => [1,2,4,3,5] => 0
[3,2,4,5,1] => [1,3,4,5,2] => [2,5,1,3,4] => [1,2,5,4,3] => 0
[3,2,5,1,4] => [1,3,5,4,2] => [3,5,2,4,1] => [1,3,2,5,4] => 1
[3,2,5,4,1] => [1,3,5,2,4] => [2,1,4,5,3] => [1,2,3,4,5] => 0
[3,4,1,2,5] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,4,1,5,2] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,4,2,1,5] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,4,2,5,1] => [1,3,2,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[3,4,5,1,2] => [1,3,5,2,4] => [2,1,4,5,3] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [1,3,5,2,4] => [2,1,4,5,3] => [1,2,3,4,5] => 0
[3,5,1,2,4] => [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,4,5] => 1
[3,5,1,4,2] => [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,4,5] => 1
[3,5,2,1,4] => [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,4,5] => 1
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation π of length n,
#{2<k≤n∣π(2)>π(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 π of length n,
#{2<k≤n∣π(2)>π(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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
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