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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000009
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [2,1] => [[1],[2]]
=> 0
[2,1] => [2,1] => [1,2] => [[1,2]]
=> 1
[1,2,3] => [1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 0
[1,3,2] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 2
[2,1,3] => [2,1,3] => [2,3,1] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [3,1,2] => [[1,2],[3]]
=> 2
[3,1,2] => [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [[1,2,3]]
=> 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 3
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 3
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 5
[2,1,3,4] => [2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 4
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[2,3,4,1] => [1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [[1,2],[3,4]]
=> 4
[2,4,3,1] => [1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 5
[3,1,2,4] => [3,1,2,4] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 4
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => [[1,3,4],[2]]
=> 3
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 4
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [[1,2],[3,4]]
=> 4
[3,4,2,1] => [1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 5
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 5
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => [[1,2,4],[3]]
=> 4
[4,2,3,1] => [4,1,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 5
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 5
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 6
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,2,4,5,3] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 4
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 7
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 6
[1,3,4,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,3,4,5,2] => [1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [[1,2],[3,4],[5]]
=> 6
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 7
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,4,2,5,3] => [1,3,2,5,4] => [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 6
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4,3,5,2] => [1,3,2,5,4] => [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 6
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,4,2] => [[1,2],[3,4],[5]]
=> 6
Description
The charge of a standard tableau.
Matching statistic: St000008
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2] => 0
[2,1] => [2,1] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [2,1] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1] => 2
[3,1,2] => [3,1,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [3,1] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,2] => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [3,1] => 3
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [3,1] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [2,1,1] => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,2,1] => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [2,2] => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [3,1] => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,2,1] => 4
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [2,1,1] => 5
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [2,2] => 2
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [1,2,1] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [1,2,1] => 4
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [1,2,1] => 4
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [2,1,1] => 5
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [3,1] => 3
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [2,1,1] => 5
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [1,2,1] => 4
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [2,1,1] => 5
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [2,1,1] => 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [3,2] => 3
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,3] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => 6
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [3,2] => 3
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => 6
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => 7
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [3,2] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => 6
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => 5
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => 6
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => 6
[3,2,5,4,7,6,1,8] => [2,1,4,3,7,6,5,8] => [2,1,4,3,7,6,5,8] => ? => ? = 15
[5,7,4,3,1,8,2,6] => [2,6,5,4,1,8,3,7] => [5,1,7,4,3,2,8,6] => ? => ? = 20
[3,2,7,8,1,4,5,6] => [3,2,6,8,1,4,5,7] => [5,2,1,6,7,3,8,4] => ? => ? = 15
[2,1,6,7,8,4,3,5] => [2,1,3,6,8,5,4,7] => [2,1,3,7,6,4,8,5] => ? => ? = 17
[5,4,2,7,6,1,3,8] => [4,3,2,7,6,1,5,8] => [6,3,2,1,7,5,4,8] => ? => ? = 17
[6,3,8,2,1,5,7,4] => [5,3,8,2,1,4,7,6] => [5,4,2,6,1,8,7,3] => ? => ? = 20
[5,3,2,8,1,6,7,4] => [4,3,2,8,1,5,7,6] => [5,3,2,1,6,8,7,4] => ? => ? = 19
[6,3,8,2,1,4,7,5] => [5,3,8,2,1,4,7,6] => [5,4,2,6,1,8,7,3] => ? => ? = 20
[1,6,8,3,2,7,5,4] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[1,5,2,8,3,7,6,4] => [1,4,2,8,3,7,6,5] => [1,3,5,2,8,7,6,4] => ? => ? = 21
[2,6,8,3,1,7,5,4] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[3,6,8,2,1,7,5,4] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[7,4,5,3,8,2,1,6] => [6,1,5,4,8,3,2,7] => [2,7,6,4,3,1,8,5] => ? => ? = 21
[4,2,6,7,8,3,1,5] => [2,1,3,6,8,5,4,7] => [2,1,3,7,6,4,8,5] => ? => ? = 17
[3,7,4,2,1,5,8,6] => [1,6,4,3,2,5,8,7] => [1,5,4,3,6,2,8,7] => ? => ? = 17
[7,4,5,8,6,2,1,3] => [5,1,4,8,7,3,2,6] => [2,7,6,3,1,8,5,4] => ? => ? = 22
[6,7,8,2,1,5,4,3] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[5,8,6,2,4,1,3,7] => [3,8,6,2,5,1,4,7] => [6,4,1,7,5,3,8,2] => ? => ? = 19
[8,1,5,4,7,6,2,3] => [8,1,4,3,7,6,2,5] => [2,7,4,3,8,6,5,1] => ? => ? = 23
[8,4,3,7,1,5,2,6] => [8,3,2,7,1,5,4,6] => [5,3,2,7,6,8,4,1] => ? => ? = 20
[4,5,2,3,1,8,7,6] => [2,5,1,4,3,8,7,6] => [3,1,5,4,2,8,7,6] => ? => ? = 21
[7,1,8,6,5,2,4,3] => [3,1,8,7,6,2,5,4] => [2,6,1,8,7,5,4,3] => ? => ? = 24
[8,2,4,5,7,3,6,1] => [8,1,2,4,7,3,6,5] => [2,3,6,4,8,7,5,1] => ? => ? = 21
[4,6,8,3,2,7,5,1] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[3,8,2,4,6,7,5,1] => [2,8,1,3,4,7,6,5] => [3,1,4,5,8,7,6,2] => ? => ? = 19
[4,6,8,2,1,7,5,3] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[8,2,6,5,7,4,1,3] => [8,1,4,3,7,6,2,5] => [2,7,4,3,8,6,5,1] => ? => ? = 23
[4,1,8,3,5,6,7,2] => [3,1,8,2,4,5,7,6] => [2,4,1,5,6,8,7,3] => ? => ? = 15
[3,1,8,2,5,6,7,4] => [3,1,8,2,4,5,7,6] => [2,4,1,5,6,8,7,3] => ? => ? = 15
[5,7,8,4,3,6,2,1] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[7,1,6,4,5,8,2,3] => [6,1,5,2,4,8,3,7] => [2,4,7,5,3,1,8,6] => ? => ? = 19
[4,7,8,2,1,6,5,3] => [1,4,8,3,2,7,6,5] => [1,5,4,2,8,7,6,3] => ? => ? = 23
[3,2,7,6,4,5,1,8] => [2,1,7,6,3,5,4,8] => [2,1,5,7,6,4,3,8] => ? => ? = 16
[5,7,4,6,2,8,3,1] => [2,5,1,4,3,8,7,6] => [3,1,5,4,2,8,7,6] => ? => ? = 21
[8,1,6,4,5,7,2,3] => [8,1,5,2,4,7,3,6] => [2,4,7,5,3,8,6,1] => ? => ? = 20
[4,6,1,2,8,7,3,5] => [3,5,1,2,8,7,4,6] => [3,4,1,7,2,8,6,5] => ? => ? = 19
[8,4,2,1,7,5,3,6] => [8,3,2,1,7,5,4,6] => [4,3,2,7,6,8,5,1] => ? => ? = 20
[7,1,6,5,4,8,2,3] => [6,1,5,4,3,8,2,7] => [2,7,5,4,3,1,8,6] => ? => ? = 21
[5,6,2,1,8,7,3,4] => [3,5,2,1,8,7,4,6] => [4,3,1,7,2,8,6,5] => ? => ? = 20
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000391
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => => ? = 0
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [2,1] => [2,1] => 1 => 1
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => [1,3,2] => 01 => 2
[2,1,3] => [2,1,3] => [2,1,3] => 10 => 1
[2,3,1] => [1,3,2] => [1,3,2] => 01 => 2
[3,1,2] => [3,1,2] => [2,3,1] => 01 => 2
[3,2,1] => [3,2,1] => [3,2,1] => 11 => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 001 => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 001 => 3
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 001 => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 011 => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 100 => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 101 => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 010 => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 001 => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 101 => 4
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 011 => 5
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 010 => 2
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 101 => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 110 => 3
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 101 => 4
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 101 => 4
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 011 => 5
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 001 => 3
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 011 => 5
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 101 => 4
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 011 => 5
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 011 => 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 111 => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 0001 => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 3
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 0010 => 3
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0110 => 5
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7
[3,2,1,6,5,8,7,4] => [3,2,1,5,4,8,7,6] => [3,2,1,5,4,8,7,6] => ? => ? = 20
[2,4,3,1,7,6,8,5] => [1,4,3,2,6,5,8,7] => [1,4,3,2,6,5,8,7] => ? => ? = 17
[4,5,6,3,2,1,8,7] => [1,2,6,5,4,3,8,7] => [1,2,6,5,4,3,8,7] => ? => ? = 19
[3,4,2,1,6,5,8,7] => [1,4,3,2,6,5,8,7] => [1,4,3,2,6,5,8,7] => ? => ? = 17
[2,4,3,1,6,5,8,7] => [1,4,3,2,6,5,8,7] => [1,4,3,2,6,5,8,7] => ? => ? = 17
[3,2,5,4,7,6,1,8] => [2,1,4,3,7,6,5,8] => [2,1,4,3,7,6,5,8] => ? => ? = 15
[2,1,4,5,8,6,7,3] => [2,1,3,4,8,5,7,6] => [2,1,3,4,6,8,7,5] => ? => ? = 14
[4,1,2,7,8,3,5,6] => [4,1,2,6,8,3,5,7] => [2,3,6,1,7,4,8,5] => ? => ? = 15
[5,6,1,2,3,8,4,7] => [4,6,1,2,3,8,5,7] => [3,4,5,1,7,2,8,6] => ? => ? = 15
[6,1,7,8,2,3,4,5] => [4,1,6,8,2,3,5,7] => [2,5,6,1,7,3,8,4] => ? => ? = 15
[4,6,5,1,3,2,8,7] => [2,6,5,1,4,3,8,7] => [4,1,6,5,3,2,8,7] => ? => ? = 20
[5,7,4,3,1,8,2,6] => [2,6,5,4,1,8,3,7] => [5,1,7,4,3,2,8,6] => ? => ? = 20
[2,1,7,6,8,4,3,5] => [2,1,6,5,8,4,3,7] => [2,1,7,6,4,3,8,5] => ? => ? = 20
[5,7,6,8,1,3,2,4] => [2,6,5,8,1,4,3,7] => [5,1,7,6,3,2,8,4] => ? => ? = 20
[5,4,7,2,1,8,3,6] => [4,3,6,2,1,8,5,7] => [5,4,2,1,7,3,8,6] => ? => ? = 18
[2,1,6,8,7,3,5,4] => [2,1,4,8,7,3,6,5] => [2,1,6,3,8,7,5,4] => ? => ? = 22
[3,7,1,8,6,5,2,4] => [2,4,1,8,7,6,3,5] => [3,1,7,2,8,6,5,4] => ? => ? = 22
[3,2,7,8,1,4,5,6] => [3,2,6,8,1,4,5,7] => [5,2,1,6,7,3,8,4] => ? => ? = 15
[6,5,2,4,8,1,3,7] => [6,5,2,4,8,1,3,7] => [6,3,7,4,2,1,8,5] => ? => ? = 20
[6,4,8,2,7,1,3,5] => [5,4,8,2,7,1,3,6] => [6,4,7,2,1,8,5,3] => ? => ? = 21
[7,5,8,2,6,1,3,4] => [5,4,8,2,7,1,3,6] => [6,4,7,2,1,8,5,3] => ? => ? = 21
[6,3,8,1,2,7,4,5] => [5,3,8,1,2,7,4,6] => [4,5,2,7,1,8,6,3] => ? => ? = 19
[8,3,4,1,6,7,2,5] => [8,1,3,2,5,7,4,6] => [2,4,3,7,5,8,6,1] => ? => ? = 19
[4,6,7,3,8,2,5,1] => [1,2,5,4,8,3,7,6] => [1,2,6,4,3,8,7,5] => ? => ? = 20
[4,6,7,3,8,2,1,5] => [1,2,6,5,8,4,3,7] => [1,2,7,6,4,3,8,5] => ? => ? = 19
[4,6,7,3,8,1,5,2] => [1,2,5,4,8,3,7,6] => [1,2,6,4,3,8,7,5] => ? => ? = 20
[4,6,7,1,8,3,5,2] => [1,3,5,2,8,4,7,6] => [1,4,2,6,3,8,7,5] => ? => ? = 19
[2,1,6,7,8,4,3,5] => [2,1,3,6,8,5,4,7] => [2,1,3,7,6,4,8,5] => ? => ? = 17
[4,6,7,2,8,3,5,1] => [1,3,5,2,8,4,7,6] => [1,4,2,6,3,8,7,5] => ? => ? = 19
[4,6,7,2,8,1,5,3] => [1,2,5,4,8,3,7,6] => [1,2,6,4,3,8,7,5] => ? => ? = 20
[4,6,7,1,8,2,5,3] => [1,3,5,2,8,4,7,6] => [1,4,2,6,3,8,7,5] => ? => ? = 19
[1,6,7,3,2,5,4,8] => [1,4,7,3,2,6,5,8] => [1,5,4,2,7,6,3,8] => ? => ? = 16
[3,7,1,5,2,8,4,6] => [3,6,1,5,2,8,4,7] => [3,5,1,7,4,2,8,6] => ? => ? = 18
[3,6,1,8,7,2,4,5] => [3,5,1,8,7,2,4,6] => [3,6,1,7,2,8,5,4] => ? => ? = 19
[4,6,8,1,7,2,3,5] => [3,5,8,1,7,2,4,6] => [4,6,1,7,2,8,5,3] => ? => ? = 19
[4,7,8,1,6,2,3,5] => [3,5,8,1,7,2,4,6] => [4,6,1,7,2,8,5,3] => ? => ? = 19
[3,7,1,8,6,2,4,5] => [3,5,1,8,7,2,4,6] => [3,6,1,7,2,8,5,4] => ? => ? = 19
[6,4,7,3,5,8,1,2] => [3,2,6,1,5,8,4,7] => [4,2,1,7,5,3,8,6] => ? => ? = 19
[5,4,2,7,6,1,3,8] => [4,3,2,7,6,1,5,8] => [6,3,2,1,7,5,4,8] => ? => ? = 17
[5,4,2,6,7,1,8,3] => [3,2,1,4,6,5,8,7] => [3,2,1,4,6,5,8,7] => ? => ? = 15
[5,3,2,6,1,7,8,4] => [3,2,1,5,4,6,8,7] => [3,2,1,5,4,6,8,7] => ? => ? = 14
[5,3,2,6,1,8,7,4] => [3,2,1,5,4,8,7,6] => [3,2,1,5,4,8,7,6] => ? => ? = 20
[5,4,2,6,8,1,7,3] => [3,2,1,5,8,4,7,6] => [3,2,1,6,4,8,7,5] => ? => ? = 20
[6,3,8,2,1,5,7,4] => [5,3,8,2,1,4,7,6] => [5,4,2,6,1,8,7,3] => ? => ? = 20
[5,3,2,8,1,6,7,4] => [4,3,2,8,1,5,7,6] => [5,3,2,1,6,8,7,4] => ? => ? = 19
[5,4,2,7,8,1,6,3] => [3,2,1,5,8,4,7,6] => [3,2,1,6,4,8,7,5] => ? => ? = 20
[5,3,2,7,1,8,6,4] => [3,2,1,5,4,8,7,6] => [3,2,1,5,4,8,7,6] => ? => ? = 20
[6,3,7,2,1,8,4,5] => [4,3,6,2,1,8,5,7] => [5,4,2,1,7,3,8,6] => ? => ? = 18
[6,3,7,2,1,4,5,8] => [5,3,7,2,1,4,6,8] => [5,4,2,6,1,7,3,8] => ? => ? = 13
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000330
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
=> 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> 3
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 5
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 4
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 2
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
=> 4
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 5
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> 2
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 4
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 4
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 4
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 5
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> 3
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> 5
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> 4
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> 5
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> 5
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 7
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 6
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 7
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 3
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 6
[5,6,7,8,4,2,3,1] => [1,2,4,8,7,3,6,5] => ?
=> ? = 21
[8,6,5,7,3,2,4,1] => [8,4,3,7,2,1,6,5] => ?
=> ? = 24
[8,5,6,7,2,3,4,1] => [8,2,4,7,1,3,6,5] => ?
=> ? = 22
[7,8,3,2,4,5,6,1] => [5,8,2,1,3,4,7,6] => ?
=> ? = 18
[5,4,3,6,7,8,1,2] => [3,2,1,4,6,8,5,7] => ?
=> ? = 15
[5,6,7,8,3,1,2,4] => [1,3,6,8,5,2,4,7] => ?
=> ? = 18
[7,6,8,5,1,2,3,4] => [5,4,8,7,1,2,3,6] => ?
=> ? = 20
[7,8,5,6,1,2,3,4] => [5,8,4,7,1,2,3,6] => ?
=> ? = 20
[8,6,5,7,1,2,3,4] => [8,5,4,7,1,2,3,6] => ?
=> ? = 20
[7,6,5,8,1,2,3,4] => [6,5,4,8,1,2,3,7] => ?
=> ? = 19
[8,5,6,3,4,1,2,7] => [8,3,6,2,5,1,4,7] => ?
=> ? = 19
[8,5,6,1,2,3,4,7] => [8,4,6,1,2,3,5,7] => ?
=> ? = 15
[6,7,5,2,3,4,1,8] => [3,7,6,1,2,5,4,8] => ?
=> ? = 17
[3,2,1,6,5,8,7,4] => [3,2,1,5,4,8,7,6] => ?
=> ? = 20
[2,4,3,1,7,6,8,5] => [1,4,3,2,6,5,8,7] => ?
=> ? = 17
[4,5,6,3,2,1,8,7] => [1,2,6,5,4,3,8,7] => ?
=> ? = 19
[3,4,2,1,6,5,8,7] => [1,4,3,2,6,5,8,7] => ?
=> ? = 17
[2,4,3,1,6,5,8,7] => [1,4,3,2,6,5,8,7] => ?
=> ? = 17
[3,2,5,4,7,6,1,8] => [2,1,4,3,7,6,5,8] => ?
=> ? = 15
[4,2,3,1,8,7,6,5] => [4,1,3,2,8,7,6,5] => ?
=> ? = 23
[8,5,4,3,2,7,6,1] => [8,4,3,2,1,7,6,5] => ?
=> ? = 24
[3,4,6,1,7,8,2,5] => [1,2,4,3,6,8,5,7] => ?
=> ? = 15
[4,1,2,7,8,3,5,6] => [4,1,2,6,8,3,5,7] => ?
=> ? = 15
[5,6,1,2,3,8,4,7] => [4,6,1,2,3,8,5,7] => ?
=> ? = 15
[5,3,2,6,1,4,8,7] => [4,3,2,6,1,5,8,7] => ?
=> ? = 18
[7,3,2,5,4,8,1,6] => [6,2,1,5,4,8,3,7] => ?
=> ? = 20
[6,8,4,3,7,1,5,2] => [4,8,3,2,7,1,6,5] => ?
=> ? = 24
[4,8,6,1,7,3,5,2] => [2,8,4,1,7,3,6,5] => ?
=> ? = 22
[2,1,7,6,8,4,3,5] => [2,1,6,5,8,4,3,7] => ?
=> ? = 20
[3,7,1,6,8,4,2,5] => [2,6,1,5,8,4,3,7] => ?
=> ? = 20
[7,3,2,6,8,4,1,5] => [6,2,1,5,8,4,3,7] => ?
=> ? = 20
[5,4,8,2,1,7,6,3] => [4,3,8,2,1,7,6,5] => ?
=> ? = 24
[3,5,1,8,2,7,6,4] => [2,4,1,8,3,7,6,5] => ?
=> ? = 22
[4,7,8,1,6,5,2,3] => [2,4,8,1,7,6,3,5] => ?
=> ? = 22
[3,7,1,8,6,5,2,4] => [2,4,1,8,7,6,3,5] => ?
=> ? = 22
[6,8,5,7,1,2,3,4] => [5,8,4,7,1,2,3,6] => ?
=> ? = 20
[7,8,2,4,1,3,5,6] => [6,8,2,4,1,3,5,7] => ?
=> ? = 16
[5,6,2,4,8,1,3,7] => [3,6,2,5,8,1,4,7] => ?
=> ? = 19
[5,7,2,4,8,1,3,6] => [3,6,2,5,8,1,4,7] => ?
=> ? = 19
[4,7,3,6,8,1,2,5] => [3,6,2,5,8,1,4,7] => ?
=> ? = 19
[8,3,1,5,6,2,4,7] => [8,2,1,4,6,3,5,7] => ?
=> ? = 16
[8,3,4,1,6,7,2,5] => [8,1,3,2,5,7,4,6] => ?
=> ? = 19
[4,6,7,3,8,2,1,5] => [1,2,6,5,8,4,3,7] => ?
=> ? = 19
[2,1,6,7,8,4,3,5] => [2,1,3,6,8,5,4,7] => ?
=> ? = 17
[4,6,7,2,8,1,3,5] => [1,4,6,3,8,2,5,7] => ?
=> ? = 17
[3,4,2,1,7,8,5,6] => [1,4,3,2,6,8,5,7] => ?
=> ? = 17
[5,6,7,8,4,1,3,2] => [1,2,4,8,7,3,6,5] => ?
=> ? = 21
[1,6,7,4,5,2,3,8] => [1,4,7,3,6,2,5,8] => ?
=> ? = 16
[6,3,8,4,5,7,1,2] => [5,1,8,2,4,7,3,6] => ?
=> ? = 20
[7,8,1,4,5,2,3,6] => [6,8,1,3,5,2,4,7] => ?
=> ? = 18
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000169
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 4
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 6
[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] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 7
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 7
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> 5
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[5,6,7,8,4,2,3,1] => [1,2,4,8,7,3,6,5] => ?
=> ?
=> ? = 21
[8,6,5,7,3,2,4,1] => [8,4,3,7,2,1,6,5] => ?
=> ?
=> ? = 24
[8,5,6,7,2,3,4,1] => [8,2,4,7,1,3,6,5] => ?
=> ?
=> ? = 22
[7,8,3,2,4,5,6,1] => [5,8,2,1,3,4,7,6] => ?
=> ?
=> ? = 18
[5,4,3,6,7,8,1,2] => [3,2,1,4,6,8,5,7] => ?
=> ?
=> ? = 15
[5,6,7,8,3,1,2,4] => [1,3,6,8,5,2,4,7] => ?
=> ?
=> ? = 18
[7,6,8,5,1,2,3,4] => [5,4,8,7,1,2,3,6] => ?
=> ?
=> ? = 20
[7,8,5,6,1,2,3,4] => [5,8,4,7,1,2,3,6] => ?
=> ?
=> ? = 20
[8,6,5,7,1,2,3,4] => [8,5,4,7,1,2,3,6] => ?
=> ?
=> ? = 20
[7,6,5,8,1,2,3,4] => [6,5,4,8,1,2,3,7] => ?
=> ?
=> ? = 19
[8,5,6,3,4,1,2,7] => [8,3,6,2,5,1,4,7] => ?
=> ?
=> ? = 19
[8,5,6,1,2,3,4,7] => [8,4,6,1,2,3,5,7] => ?
=> ?
=> ? = 15
[6,7,5,2,3,4,1,8] => [3,7,6,1,2,5,4,8] => ?
=> ?
=> ? = 17
[3,2,1,6,5,8,7,4] => [3,2,1,5,4,8,7,6] => ?
=> ?
=> ? = 20
[2,4,3,1,7,6,8,5] => [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 17
[4,5,6,3,2,1,8,7] => [1,2,6,5,4,3,8,7] => ?
=> ?
=> ? = 19
[3,4,2,1,6,5,8,7] => [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 17
[2,4,3,1,6,5,8,7] => [1,4,3,2,6,5,8,7] => ?
=> ?
=> ? = 17
[3,2,5,4,7,6,1,8] => [2,1,4,3,7,6,5,8] => ?
=> ?
=> ? = 15
[4,2,3,1,8,7,6,5] => [4,1,3,2,8,7,6,5] => ?
=> ?
=> ? = 23
[8,5,4,3,2,7,6,1] => [8,4,3,2,1,7,6,5] => ?
=> ?
=> ? = 24
[3,4,6,1,7,8,2,5] => [1,2,4,3,6,8,5,7] => ?
=> ?
=> ? = 15
[4,1,2,7,8,3,5,6] => [4,1,2,6,8,3,5,7] => ?
=> ?
=> ? = 15
[5,6,1,2,3,8,4,7] => [4,6,1,2,3,8,5,7] => ?
=> ?
=> ? = 15
[5,3,2,6,1,4,8,7] => [4,3,2,6,1,5,8,7] => ?
=> ?
=> ? = 18
[7,3,2,5,4,8,1,6] => [6,2,1,5,4,8,3,7] => ?
=> ?
=> ? = 20
[6,8,4,3,7,1,5,2] => [4,8,3,2,7,1,6,5] => ?
=> ?
=> ? = 24
[4,8,6,1,7,3,5,2] => [2,8,4,1,7,3,6,5] => ?
=> ?
=> ? = 22
[2,1,7,6,8,4,3,5] => [2,1,6,5,8,4,3,7] => ?
=> ?
=> ? = 20
[3,7,1,6,8,4,2,5] => [2,6,1,5,8,4,3,7] => ?
=> ?
=> ? = 20
[7,3,2,6,8,4,1,5] => [6,2,1,5,8,4,3,7] => ?
=> ?
=> ? = 20
[5,4,8,2,1,7,6,3] => [4,3,8,2,1,7,6,5] => ?
=> ?
=> ? = 24
[3,5,1,8,2,7,6,4] => [2,4,1,8,3,7,6,5] => ?
=> ?
=> ? = 22
[4,7,8,1,6,5,2,3] => [2,4,8,1,7,6,3,5] => ?
=> ?
=> ? = 22
[3,7,1,8,6,5,2,4] => [2,4,1,8,7,6,3,5] => ?
=> ?
=> ? = 22
[6,8,5,7,1,2,3,4] => [5,8,4,7,1,2,3,6] => ?
=> ?
=> ? = 20
[7,8,2,4,1,3,5,6] => [6,8,2,4,1,3,5,7] => ?
=> ?
=> ? = 16
[5,6,2,4,8,1,3,7] => [3,6,2,5,8,1,4,7] => ?
=> ?
=> ? = 19
[5,7,2,4,8,1,3,6] => [3,6,2,5,8,1,4,7] => ?
=> ?
=> ? = 19
[4,7,3,6,8,1,2,5] => [3,6,2,5,8,1,4,7] => ?
=> ?
=> ? = 19
[8,3,1,5,6,2,4,7] => [8,2,1,4,6,3,5,7] => ?
=> ?
=> ? = 16
[8,3,4,1,6,7,2,5] => [8,1,3,2,5,7,4,6] => ?
=> ?
=> ? = 19
[4,6,7,3,8,2,1,5] => [1,2,6,5,8,4,3,7] => ?
=> ?
=> ? = 19
[2,1,6,7,8,4,3,5] => [2,1,3,6,8,5,4,7] => ?
=> ?
=> ? = 17
[4,6,7,2,8,1,3,5] => [1,4,6,3,8,2,5,7] => ?
=> ?
=> ? = 17
[3,4,2,1,7,8,5,6] => [1,4,3,2,6,8,5,7] => ?
=> ?
=> ? = 17
[5,6,7,8,4,1,3,2] => [1,2,4,8,7,3,6,5] => ?
=> ?
=> ? = 21
[1,6,7,4,5,2,3,8] => [1,4,7,3,6,2,5,8] => ?
=> ?
=> ? = 16
[6,3,8,4,5,7,1,2] => [5,1,8,2,4,7,3,6] => ?
=> ?
=> ? = 20
[7,8,1,4,5,2,3,6] => [6,8,1,3,5,2,4,7] => ?
=> ?
=> ? = 18
Description
The cocharge of a standard tableau.
The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm:
1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$.
2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling.
3) The cocharge is defined as the sum of the superscript labels on the letters.
Matching statistic: St001759
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 72%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001759: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 72%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [3,1,2] => [2,3,1] => 2
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 3
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => [3,2,4,1] => 4
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 5
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [2,1,4,3] => [4,2,1,3] => [3,2,4,1] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [2,1,4,3] => [4,2,1,3] => [3,2,4,1] => 4
[3,4,1,2] => [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 4
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 5
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 5
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 4
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 5
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[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] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 3
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [3,4,5,2,1] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => [3,4,2,5,1] => 6
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 3
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => [2,3,1,5,4] => 6
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => [3,4,5,2,1] => 7
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [5,3,1,2,4] => [3,4,2,5,1] => 6
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 5
[1,4,3,5,2] => [1,3,2,5,4] => [5,3,1,2,4] => [3,4,2,5,1] => 6
[1,4,5,2,3] => [1,3,5,2,4] => [3,1,2,5,4] => [2,3,1,5,4] => 6
[1,2,5,4,7,3,6] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [6,1,5,2,3,4,7] => [2,4,5,6,3,1,7] => ? = 9
[1,2,6,4,3,5,7] => [1,2,6,4,3,5,7] => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 8
[1,2,6,4,5,3,7] => [1,2,6,3,5,4,7] => [6,1,5,2,3,4,7] => [2,4,5,6,3,1,7] => ? = 9
[1,2,6,4,7,3,5] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,2,6,5,3,4,7] => [1,2,6,5,3,4,7] => [1,6,5,2,3,4,7] => [1,4,5,6,3,2,7] => ? = 9
[1,2,6,5,7,3,4] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,3,2,5,7,4,6] => [1,3,2,5,7,4,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 12
[1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => ? = 7
[1,3,2,6,7,4,5] => [1,3,2,5,7,4,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 12
[1,3,2,7,5,4,6] => [1,3,2,7,5,4,6] => [5,7,3,1,2,4,6] => [4,5,3,6,1,7,2] => ? = 12
[1,3,2,7,6,4,5] => [1,3,2,7,6,4,5] => [3,7,6,1,2,4,5] => [4,5,1,6,7,3,2] => ? = 13
[1,3,5,4,7,2,6] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,3,6,2,5,4,7] => [1,3,6,2,5,4,7] => [6,3,1,2,5,4,7] => [3,4,2,6,5,1,7] => ? = 11
[1,3,6,4,2,5,7] => [1,2,6,4,3,5,7] => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 8
[1,3,6,4,5,2,7] => [1,2,6,3,5,4,7] => [6,1,5,2,3,4,7] => [2,4,5,6,3,1,7] => ? = 9
[1,3,6,4,7,2,5] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,3,6,5,2,4,7] => [1,3,6,5,2,4,7] => [3,6,1,2,5,4,7] => [3,4,1,6,5,2,7] => ? = 11
[1,3,6,5,7,2,4] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,3,7,2,6,4,5] => [1,3,7,2,6,4,5] => [3,7,1,2,4,6,5] => [3,4,1,5,7,6,2] => ? = 13
[1,3,7,5,2,4,6] => [1,3,7,5,2,4,6] => [3,1,7,2,5,4,6] => [2,4,1,6,5,7,3] => ? = 12
[1,3,7,6,2,4,5] => [1,3,7,6,2,4,5] => [3,1,7,2,4,6,5] => [2,4,1,5,7,6,3] => ? = 13
[1,3,7,6,2,5,4] => [1,3,7,6,2,5,4] => [7,3,6,1,2,5,4] => [4,5,2,7,6,3,1] => ? = 17
[1,3,7,6,5,2,4] => [1,3,7,6,5,2,4] => [3,7,6,1,2,5,4] => [4,5,1,7,6,3,2] => ? = 17
[1,4,2,3,6,5,7] => [1,4,2,3,6,5,7] => [6,1,4,2,3,5,7] => [2,4,5,3,6,1,7] => ? = 8
[1,4,2,5,7,3,6] => [1,3,2,5,7,4,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 12
[1,4,2,6,7,3,5] => [1,3,2,5,7,4,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 12
[1,4,2,7,5,3,6] => [1,3,2,7,5,4,6] => [5,7,3,1,2,4,6] => [4,5,3,6,1,7,2] => ? = 12
[1,4,2,7,6,3,5] => [1,4,2,7,6,3,5] => [4,7,1,6,2,3,5] => [3,5,6,1,7,4,2] => ? = 14
[1,4,3,2,7,5,6] => [1,4,3,2,7,5,6] => [4,7,3,1,2,5,6] => [4,5,3,1,6,7,2] => ? = 11
[1,4,3,5,7,2,6] => [1,3,2,5,7,4,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 12
[1,4,3,6,2,5,7] => [1,4,3,6,2,5,7] => [4,3,1,2,6,5,7] => [3,4,2,1,6,5,7] => ? = 10
[1,4,3,6,7,2,5] => [1,3,2,5,7,4,6] => [5,3,1,2,4,7,6] => [3,4,2,5,1,7,6] => ? = 12
[1,4,3,7,2,5,6] => [1,4,3,7,2,5,6] => [4,3,1,2,5,7,6] => [3,4,2,1,5,7,6] => ? = 11
[1,4,3,7,5,2,6] => [1,3,2,7,5,4,6] => [5,7,3,1,2,4,6] => [4,5,3,6,1,7,2] => ? = 12
[1,4,3,7,6,2,5] => [1,4,3,7,6,2,5] => [4,7,3,1,2,6,5] => [4,5,3,1,7,6,2] => ? = 16
[1,4,5,3,7,2,6] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,4,6,2,5,3,7] => [1,3,6,2,5,4,7] => [6,3,1,2,5,4,7] => [3,4,2,6,5,1,7] => ? = 11
[1,4,6,3,2,5,7] => [1,4,6,3,2,5,7] => [4,1,6,3,2,5,7] => [2,5,4,1,6,3,7] => ? = 10
[1,4,6,3,5,2,7] => [1,3,6,2,5,4,7] => [6,3,1,2,5,4,7] => [3,4,2,6,5,1,7] => ? = 11
[1,4,6,3,7,2,5] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,4,6,5,2,3,7] => [1,3,6,5,2,4,7] => [3,6,1,2,5,4,7] => [3,4,1,6,5,2,7] => ? = 11
[1,4,6,5,7,2,3] => [1,2,5,4,7,3,6] => [5,4,1,2,3,7,6] => [3,4,5,2,1,7,6] => ? = 13
[1,4,7,3,2,5,6] => [1,4,7,3,2,5,6] => [4,1,3,7,2,5,6] => [2,5,3,1,6,7,4] => ? = 11
[1,4,7,3,2,6,5] => [1,4,7,3,2,6,5] => [7,4,1,6,3,2,5] => [3,6,5,2,7,4,1] => ? = 16
[1,4,7,3,6,2,5] => [1,4,7,3,6,2,5] => [4,3,1,2,7,6,5] => [3,4,2,1,7,6,5] => ? = 16
[1,4,7,5,2,3,6] => [1,3,7,5,2,4,6] => [3,1,7,2,5,4,6] => [2,4,1,6,5,7,3] => ? = 12
[1,4,7,6,2,5,3] => [1,3,7,6,2,5,4] => [7,3,6,1,2,5,4] => [4,5,2,7,6,3,1] => ? = 17
[1,4,7,6,3,2,5] => [1,4,7,6,3,2,5] => [4,7,1,6,3,2,5] => [3,6,5,1,7,4,2] => ? = 16
[1,4,7,6,3,5,2] => [1,3,7,6,2,5,4] => [7,3,6,1,2,5,4] => [4,5,2,7,6,3,1] => ? = 17
Description
The Rajchgot index of a permutation.
The '''Rajchgot index''' of a permutation $\sigma$ is the degree of the ''Grothendieck polynomial'' of $\sigma$. This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index [[St000004]] of the permutations smaller than or equal to $\sigma$ in the right ''weak Bruhat order''.
Matching statistic: St000018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [2,3,1] => [2,3,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 4
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 4
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 4
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [4,2,3,1] => 5
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 4
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [4,2,3,1] => 5
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[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] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,5,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 6
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [4,5,1,2,3] => 6
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 6
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 5
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 6
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [4,5,1,2,3] => 6
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 9
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,6,4] => ? = 11
[1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [6,5,1,2,3,7,4] => ? = 10
[1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,6,4] => ? = 11
[1,2,3,7,6,4,5] => [1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [6,7,1,2,3,5,4] => ? = 11
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [6,4,1,2,3,5,7] => ? = 8
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [6,4,1,2,3,7,5] => ? = 9
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,1,2,3,5] => ? = 14
[1,2,4,6,5,3,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 9
[1,2,4,7,5,3,6] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [6,5,1,2,3,7,4] => ? = 10
[1,2,4,7,5,6,3] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,6,4] => ? = 11
[1,2,4,7,6,3,5] => [1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [6,7,5,1,2,3,4] => ? = 14
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [4,1,2,5,3,6,7] => ? = 4
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [7,4,1,2,5,3,6] => ? = 10
[1,2,5,3,6,4,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [6,4,1,2,3,5,7] => ? = 8
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [6,4,1,2,7,3,5] => ? = 10
[1,2,5,3,7,6,4] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,1,2,3,5] => ? = 14
[1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => ? = 7
[1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [7,5,4,1,2,3,6] => ? = 13
[1,2,5,4,6,3,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [6,4,1,2,3,5,7] => ? = 8
[1,2,5,4,7,3,6] => [1,2,5,4,7,3,6] => [1,2,6,4,3,7,5] => [6,7,4,1,2,3,5] => ? = 13
[1,2,5,4,7,6,3] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,1,2,3,5] => ? = 14
[1,2,5,6,4,3,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 9
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [1,2,5,6,3,7,4] => [5,6,1,2,7,3,4] => ? = 10
[1,2,5,7,4,3,6] => [1,2,5,7,4,3,6] => [1,2,6,5,3,7,4] => [6,5,7,1,2,3,4] => ? = 13
[1,2,5,7,6,3,4] => [1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [6,7,5,1,2,3,4] => ? = 14
[1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => ? = 5
[1,2,6,3,4,7,5] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [7,4,1,2,5,3,6] => ? = 10
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [1,2,4,6,5,3,7] => [6,4,1,2,5,3,7] => ? = 9
[1,2,6,3,5,7,4] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [7,4,1,2,5,3,6] => ? = 10
[1,2,6,3,7,4,5] => [1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [6,4,1,2,7,3,5] => ? = 10
[1,2,6,3,7,5,4] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,1,2,3,5] => ? = 14
[1,2,6,4,3,5,7] => [1,2,6,4,3,5,7] => [1,2,5,4,6,3,7] => [5,4,1,2,6,3,7] => ? = 8
[1,2,6,4,3,7,5] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [7,5,4,1,2,3,6] => ? = 13
[1,2,6,4,5,3,7] => [1,2,6,3,5,4,7] => [1,2,4,6,5,3,7] => [6,4,1,2,5,3,7] => ? = 9
[1,2,6,4,5,7,3] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [7,4,1,2,5,3,6] => ? = 10
[1,2,6,4,7,3,5] => [1,2,5,4,7,3,6] => [1,2,6,4,3,7,5] => [6,7,4,1,2,3,5] => ? = 13
[1,2,6,4,7,5,3] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,1,2,3,5] => ? = 14
[1,2,6,5,3,4,7] => [1,2,6,5,3,4,7] => [1,2,5,6,4,3,7] => [5,6,1,2,4,3,7] => ? = 9
[1,2,6,5,3,7,4] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [7,5,4,1,2,3,6] => ? = 13
[1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 12
[1,2,6,5,4,7,3] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [7,5,4,1,2,3,6] => ? = 13
[1,2,6,5,7,3,4] => [1,2,5,4,7,3,6] => [1,2,6,4,3,7,5] => [6,7,4,1,2,3,5] => ? = 13
[1,2,6,5,7,4,3] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,6,4,1,2,3,5] => ? = 14
[1,2,6,7,3,4,5] => [1,2,5,7,3,4,6] => [1,2,5,6,3,7,4] => [5,6,1,2,7,3,4] => ? = 10
[1,2,6,7,4,3,5] => [1,2,5,7,4,3,6] => [1,2,6,5,3,7,4] => [6,5,7,1,2,3,4] => ? = 13
[1,2,6,7,5,3,4] => [1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [6,7,5,1,2,3,4] => ? = 14
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => [4,1,2,5,6,7,3] => ? = 6
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [1,2,4,5,7,6,3] => [7,4,1,2,5,6,3] => ? = 11
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [1,2,4,6,5,7,3] => [6,4,1,2,5,7,3] => ? = 10
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000446
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 55%
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [2,3,1] => [2,1,3] => 2
[3,1,2] => [3,1,2] => [2,1,3] => [2,3,1] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 3
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 3
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 3
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 5
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 4
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 4
[2,4,3,1] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 5
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 2
[3,1,4,2] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 4
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 3
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 4
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 4
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 5
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 3
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 5
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 4
[4,2,3,1] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 5
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 5
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 3
[1,2,4,5,3] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 6
[1,3,4,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 3
[1,3,4,5,2] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 6
[1,3,5,4,2] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 7
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 6
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 6
[1,4,5,2,3] => [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 6
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 6
[1,2,3,4,6,7,5] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 6
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => ? = 6
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 11
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,3,5,6,7,4] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 6
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 10
[1,2,3,5,7,6,4] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 11
[1,2,3,6,4,7,5] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,3,6,5,7,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 10
[1,2,3,6,7,5,4] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 11
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => ? = 6
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 11
[1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 10
[1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 11
[1,2,3,7,6,4,5] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => ? = 11
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 15
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 9
[1,2,4,3,6,7,5] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 9
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => ? = 9
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 14
[1,2,4,5,3,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,4,5,6,7,3] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 6
[1,2,4,5,7,3,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 10
[1,2,4,5,7,6,3] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 11
[1,2,4,6,3,7,5] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,4,6,5,7,3] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,4,6,7,3,5] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 10
[1,2,4,6,7,5,3] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 11
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [2,3,5,1,4,6,7] => ? = 9
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [3,2,5,1,4,6,7] => ? = 14
[1,2,4,7,5,3,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => ? = 10
[1,2,4,7,5,6,3] => [1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => ? = 11
[1,2,4,7,6,3,5] => [1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [3,5,2,1,4,6,7] => ? = 14
[1,2,4,7,6,5,3] => [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 15
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => ? = 10
[1,2,5,3,6,7,4] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 9
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => ? = 10
[1,2,5,3,7,6,4] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 14
[1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [2,1,5,4,3,6,7] => ? = 13
[1,2,5,4,6,7,3] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => ? = 9
[1,2,5,4,7,3,6] => [1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [2,5,1,4,3,6,7] => ? = 13
[1,2,5,4,7,6,3] => [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => ? = 14
[1,2,5,6,3,7,4] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,5,6,4,7,3] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => ? = 10
[1,2,5,6,7,3,4] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => ? = 10
[1,2,5,6,7,4,3] => [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => ? = 11
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [2,4,5,1,3,6,7] => ? = 10
[1,2,5,7,3,6,4] => [1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [3,2,5,1,4,6,7] => ? = 14
Description
The disorder of a permutation.
Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The '''disorder''' of $\pi$ is defined to be the number of times a position was not removed in this process.
For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.
Matching statistic: St000833
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Mp00066: Permutations —inverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => [3,1,2,4] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 4
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 3
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 4
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => [2,1,3,4] => 3
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => [4,2,1,3] => 5
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 4
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => [4,2,1,3] => 5
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [3,2,1,4] => 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[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] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 3
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,3,5] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 6
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => 3
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => [4,1,5,2,3] => 6
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => [1,3,2,4,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 6
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [3,1,5,2,4] => 6
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => [4,1,5,2,3] => 6
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 6
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 11
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,3,5,6,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 5
[1,2,3,5,6,7,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 10
[1,2,3,5,7,6,4] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 11
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 5
[1,2,3,6,4,7,5] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 9
[1,2,3,6,5,7,4] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 10
[1,2,3,6,7,5,4] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 11
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 6
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 11
[1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [6,1,5,2,3,4,7] => ? = 10
[1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 11
[1,2,3,7,6,4,5] => [1,2,3,7,6,4,5] => [1,2,3,6,7,5,4] => [6,5,1,2,3,4,7] => ? = 11
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 15
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 9
[1,2,4,3,6,7,5] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 9
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [4,1,2,6,3,5,7] => ? = 9
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 14
[1,2,4,5,3,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,4,5,6,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 5
[1,2,4,5,6,7,3] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 6
[1,2,4,5,7,3,6] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 10
[1,2,4,5,7,6,3] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 11
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,5,2,6,3,4,7] => ? = 8
[1,2,4,6,3,7,5] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,4,6,5,3,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 9
[1,2,4,6,5,7,3] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 10
[1,2,4,6,7,3,5] => [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 10
[1,2,4,6,7,5,3] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 11
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [5,1,2,6,3,4,7] => ? = 9
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [1,2,5,3,7,6,4] => [7,5,1,6,2,3,4] => ? = 14
[1,2,4,7,5,3,6] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => [6,1,5,2,3,4,7] => ? = 10
[1,2,4,7,5,6,3] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 11
[1,2,4,7,6,3,5] => [1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [6,5,1,7,2,3,4] => ? = 14
[1,2,4,7,6,5,3] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 15
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => ? = 10
[1,2,5,3,6,7,4] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 9
[1,2,5,3,7,4,6] => [1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => ? = 10
[1,2,5,3,7,6,4] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 14
[1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [5,1,7,4,2,3,6] => ? = 13
[1,2,5,4,6,7,3] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 9
Description
The comajor index of a permutation.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St000305
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 55%
St000305: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 55%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,2,4,3] => 3
[1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [1,4,3,2] => 5
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [1,3,2,4] => 2
[2,3,4,1] => [1,2,4,3] => 3
[2,4,1,3] => [2,4,1,3] => 4
[2,4,3,1] => [1,4,3,2] => 5
[3,1,2,4] => [3,1,2,4] => 2
[3,1,4,2] => [2,1,4,3] => 4
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [2,1,4,3] => 4
[3,4,1,2] => [2,4,1,3] => 4
[3,4,2,1] => [1,4,3,2] => 5
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 5
[4,2,1,3] => [4,2,1,3] => 4
[4,2,3,1] => [4,1,3,2] => 5
[4,3,1,2] => [4,3,1,2] => 5
[4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => [1,2,3,5,4] => 4
[1,2,5,3,4] => [1,2,5,3,4] => 4
[1,2,5,4,3] => [1,2,5,4,3] => 7
[1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 6
[1,3,4,2,5] => [1,2,4,3,5] => 3
[1,3,4,5,2] => [1,2,3,5,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => 6
[1,3,5,4,2] => [1,2,5,4,3] => 7
[1,4,2,3,5] => [1,4,2,3,5] => 3
[1,4,2,5,3] => [1,3,2,5,4] => 6
[1,4,3,2,5] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [1,3,2,5,4] => 6
[1,4,5,2,3] => [1,3,5,2,4] => 6
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,5,7,6] => ? = 6
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 6
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 11
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 4
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 10
[1,2,3,5,6,4,7] => [1,2,3,4,6,5,7] => ? = 5
[1,2,3,5,6,7,4] => [1,2,3,4,5,7,6] => ? = 6
[1,2,3,5,7,4,6] => [1,2,3,5,7,4,6] => ? = 10
[1,2,3,5,7,6,4] => [1,2,3,4,7,6,5] => ? = 11
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 5
[1,2,3,6,4,7,5] => [1,2,3,5,4,7,6] => ? = 10
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 9
[1,2,3,6,5,7,4] => [1,2,3,5,4,7,6] => ? = 10
[1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => ? = 10
[1,2,3,6,7,5,4] => [1,2,3,4,7,6,5] => ? = 11
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 6
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => ? = 11
[1,2,3,7,5,4,6] => [1,2,3,7,5,4,6] => ? = 10
[1,2,3,7,5,6,4] => [1,2,3,7,4,6,5] => ? = 11
[1,2,3,7,6,4,5] => [1,2,3,7,6,4,5] => ? = 11
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 15
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 9
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 8
[1,2,4,3,6,7,5] => [1,2,4,3,5,7,6] => ? = 9
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 9
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 14
[1,2,4,5,3,6,7] => [1,2,3,5,4,6,7] => ? = 4
[1,2,4,5,3,7,6] => [1,2,3,5,4,7,6] => ? = 10
[1,2,4,5,6,3,7] => [1,2,3,4,6,5,7] => ? = 5
[1,2,4,5,6,7,3] => [1,2,3,4,5,7,6] => ? = 6
[1,2,4,5,7,3,6] => [1,2,3,5,7,4,6] => ? = 10
[1,2,4,5,7,6,3] => [1,2,3,4,7,6,5] => ? = 11
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => ? = 8
[1,2,4,6,3,7,5] => [1,2,3,5,4,7,6] => ? = 10
[1,2,4,6,5,3,7] => [1,2,3,6,5,4,7] => ? = 9
[1,2,4,6,5,7,3] => [1,2,3,5,4,7,6] => ? = 10
[1,2,4,6,7,3,5] => [1,2,3,5,7,4,6] => ? = 10
[1,2,4,6,7,5,3] => [1,2,3,4,7,6,5] => ? = 11
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => ? = 9
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => ? = 14
[1,2,4,7,5,3,6] => [1,2,3,7,5,4,6] => ? = 10
[1,2,4,7,5,6,3] => [1,2,3,7,4,6,5] => ? = 11
[1,2,4,7,6,3,5] => [1,2,4,7,6,3,5] => ? = 14
[1,2,4,7,6,5,3] => [1,2,3,7,6,5,4] => ? = 15
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 4
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ? = 10
Description
The inverse major index of a permutation.
This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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