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Your data matches 61 different statistics following compositions of up to 3 maps.
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Matching statistic: St000884
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[2,1,3] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,2,3] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 1
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,2,4,1] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[3,4,2,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1
[4,3,1,2] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,3,2] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [5,3,1,4,2] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [1,4,5,2,3] => 1
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000035
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [1,3,2] => 1
[2,1,3] => [2,3,1] => [2,3,1] => [2,1,3] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,2,1] => [1,2,3] => [1,3,2] => [3,1,2] => 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 1
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,2,4,1] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[3,4,2,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1
[4,3,1,2] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,3,2] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [5,3,1,4,2] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,4,2,3,5] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => [1,4,2,5,3] => 2
[1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [1,4,5,2,3] => 1
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => [1,2,3,4,6,7,5] => ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,6,5,4,1,3,2] => [1,2,3,4,7,5,6] => ? = 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [7,6,5,4,1,3,2] => [1,2,3,4,7,5,6] => ? = 1
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ? = 2
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => [1,2,3,5,6,4,7] => ? = 1
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => [1,2,3,5,6,7,4] => ? = 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [7,6,5,3,1,4,2] => [1,2,3,5,7,4,6] => ? = 1
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [7,6,5,3,1,4,2] => [1,2,3,5,7,4,6] => ? = 1
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,6,5,2,4,3,1] => [1,2,3,6,4,5,7] => ? = 1
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [7,6,5,2,4,3,1] => [1,2,3,6,4,5,7] => ? = 1
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [7,6,5,2,1,4,3] => [1,2,3,6,7,4,5] => ? = 1
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => [7,6,5,2,1,4,3] => [1,2,3,6,7,4,5] => ? = 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,5,1,4,3,2] => [1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [7,6,5,1,4,3,2] => [1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [7,6,5,1,4,3,2] => [1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [7,6,5,1,4,3,2] => [1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [7,6,5,1,4,3,2] => [1,2,3,7,4,5,6] => ? = 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [7,6,5,1,4,3,2] => [1,2,3,7,4,5,6] => ? = 1
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => [1,2,4,3,5,7,6] => ? = 2
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => [1,2,4,3,6,5,7] => ? = 2
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [7,6,4,5,2,1,3] => [1,2,4,3,6,7,5] => ? = 2
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [7,6,4,5,1,3,2] => [1,2,4,3,7,5,6] => ? = 2
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [7,6,4,5,1,3,2] => [1,2,4,3,7,5,6] => ? = 2
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => [1,2,4,5,3,6,7] => ? = 1
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [7,6,4,3,5,1,2] => [1,2,4,5,3,7,6] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [7,6,4,3,2,5,1] => [1,2,4,5,6,3,7] => ? = 1
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => [1,2,4,5,6,7,3] => ? = 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [7,6,4,3,1,5,2] => [1,2,4,5,7,3,6] => ? = 1
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [7,6,4,3,1,5,2] => [1,2,4,5,7,3,6] => ? = 1
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,6,4,2,5,3,1] => [1,2,4,6,3,5,7] => ? = 1
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [7,6,4,2,5,1,3] => [1,2,4,6,3,7,5] => ? = 2
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [7,6,4,2,5,3,1] => [1,2,4,6,3,5,7] => ? = 1
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [7,6,4,2,5,1,3] => [1,2,4,6,3,7,5] => ? = 2
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [7,6,4,2,1,5,3] => [1,2,4,6,7,3,5] => ? = 1
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => [7,6,4,2,1,5,3] => [1,2,4,6,7,3,5] => ? = 1
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [7,6,4,1,5,3,2] => [1,2,4,7,3,5,6] => ? = 1
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,6,3,5,4,2,1] => [1,2,5,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [7,6,3,5,4,1,2] => [1,2,5,3,4,7,6] => ? = 2
[1,2,5,3,6,4,7] => [7,6,3,5,2,4,1] => [7,6,3,5,2,4,1] => [1,2,5,3,6,4,7] => ? = 2
[1,2,5,3,6,7,4] => [7,6,3,5,2,1,4] => [7,6,3,5,2,1,4] => [1,2,5,3,6,7,4] => ? = 2
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => [7,6,3,5,1,4,2] => [1,2,5,3,7,4,6] => ? = 2
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => [7,6,3,5,1,4,2] => [1,2,5,3,7,4,6] => ? = 2
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,3,5,4,2,1] => [1,2,5,3,4,6,7] => ? = 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000390
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => => ? = 0
[1,2] => [1,0,1,0]
=> [1,2] => 0 => 0
[2,1] => [1,1,0,0]
=> [2,1] => 1 => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 1
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 101 => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 101 => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 011 => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 011 => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0011 => 1
[7,8,5,4,6,3,2,1] => ?
=> ? => ? => ? = 1
[7,8,4,5,6,3,2,1] => ?
=> ? => ? => ? = 1
[7,8,4,3,5,6,2,1] => ?
=> ? => ? => ? = 1
[8,4,5,6,3,7,2,1] => ?
=> ? => ? => ? = 1
[8,5,6,3,4,7,2,1] => ?
=> ? => ? => ? = 1
[8,4,5,3,6,7,2,1] => ?
=> ? => ? => ? = 1
[7,5,6,3,4,8,2,1] => ?
=> ? => ? => ? = 2
[6,7,8,5,4,2,3,1] => ?
=> ? => ? => ? = 1
[7,8,5,6,3,2,4,1] => ?
=> ? => ? => ? = 1
[8,7,6,4,2,3,5,1] => ?
=> ? => ? => ? = 1
[8,6,7,4,2,3,5,1] => ?
=> ? => ? => ? = 1
[8,7,4,5,3,2,6,1] => ?
=> ? => ? => ? = 1
[7,8,4,5,3,2,6,1] => ?
=> ? => ? => ? = 1
[7,8,5,3,2,4,6,1] => ?
=> ? => ? => ? = 1
[8,7,3,2,4,5,6,1] => ?
=> ? => ? => ? = 1
[8,6,4,5,2,3,7,1] => ?
=> ? => ? => ? = 1
[8,6,3,4,2,5,7,1] => ?
=> ? => ? => ? = 1
[7,5,4,6,3,2,8,1] => ?
=> ? => ? => ? = 2
[6,7,3,4,5,2,8,1] => ?
=> ? => ? => ? = 2
[7,6,4,3,2,5,8,1] => ?
=> ? => ? => ? = 2
[7,6,3,4,2,5,8,1] => ?
=> ? => ? => ? = 2
[6,7,3,4,2,5,8,1] => ?
=> ? => ? => ? = 2
[6,7,4,2,3,5,8,1] => ?
=> ? => ? => ? = 2
[7,4,5,3,2,6,8,1] => ?
=> ? => ? => ? = 2
[7,8,5,4,6,3,1,2] => ?
=> ? => ? => ? = 1
[8,7,6,4,3,5,1,2] => ?
=> ? => ? => ? = 1
[7,8,6,4,3,5,1,2] => ?
=> ? => ? => ? = 1
[8,6,7,4,3,5,1,2] => ?
=> ? => ? => ? = 1
[8,7,5,4,3,6,1,2] => ?
=> ? => ? => ? = 1
[8,6,3,4,5,7,1,2] => ?
=> ? => ? => ? = 1
[8,4,5,3,6,7,1,2] => ?
=> ? => ? => ? = 1
[7,4,5,6,3,8,1,2] => ?
=> ? => ? => ? = 2
[8,6,5,7,4,2,1,3] => ?
=> ? => ? => ? = 1
[8,5,6,7,4,2,1,3] => ?
=> ? => ? => ? = 1
[7,8,6,4,5,2,1,3] => ?
=> ? => ? => ? = 1
[8,6,7,4,5,2,1,3] => ?
=> ? => ? => ? = 1
[8,7,5,4,6,2,1,3] => ?
=> ? => ? => ? = 1
[8,6,5,4,7,2,1,3] => ?
=> ? => ? => ? = 1
[8,6,4,5,7,2,1,3] => ?
=> ? => ? => ? = 1
[8,6,7,5,4,1,2,3] => ?
=> ? => ? => ? = 1
[8,7,5,6,4,1,2,3] => ?
=> ? => ? => ? = 1
[7,8,6,4,5,1,2,3] => ?
=> ? => ? => ? = 1
[6,7,8,5,3,1,2,4] => ?
=> ? => ? => ? = 1
[8,5,6,7,3,1,2,4] => ?
=> ? => ? => ? = 1
[8,6,7,3,4,2,1,5] => ?
=> ? => ? => ? = 1
[7,8,6,4,2,3,1,5] => ?
=> ? => ? => ? = 1
[8,6,7,4,2,3,1,5] => ?
=> ? => ? => ? = 1
[8,6,7,4,3,1,2,5] => ?
=> ? => ? => ? = 1
[8,6,7,4,1,2,3,5] => ?
=> ? => ? => ? = 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001280
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 0
[1,2] => [1,2] => [1,1]
=> 0
[2,1] => [2,1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1,3] => [2,1]
=> 1
[2,3,1] => [3,2,1] => [2,1]
=> 1
[3,1,2] => [2,3,1] => [3]
=> 1
[3,2,1] => [3,1,2] => [3]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[1,4,2,3] => [1,3,4,2] => [3,1]
=> 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> 1
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 1
[2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 1
[2,4,1,3] => [3,2,4,1] => [3,1]
=> 1
[2,4,3,1] => [4,2,1,3] => [3,1]
=> 1
[3,1,2,4] => [2,3,1,4] => [3,1]
=> 1
[3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[3,2,1,4] => [3,1,2,4] => [3,1]
=> 1
[3,2,4,1] => [4,3,2,1] => [2,2]
=> 2
[3,4,1,2] => [2,4,3,1] => [3,1]
=> 1
[3,4,2,1] => [4,1,3,2] => [3,1]
=> 1
[4,1,2,3] => [2,3,4,1] => [4]
=> 1
[4,1,3,2] => [3,4,2,1] => [4]
=> 1
[4,2,1,3] => [3,1,4,2] => [4]
=> 1
[4,2,3,1] => [4,3,1,2] => [4]
=> 1
[4,3,1,2] => [2,4,1,3] => [4]
=> 1
[4,3,2,1] => [4,1,2,3] => [4]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,2,5,3,4] => [1,2,4,5,3] => [3,1,1]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [3,1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 1
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 1
[1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 1
[1,3,5,4,2] => [1,5,3,2,4] => [3,1,1]
=> 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,1,1]
=> 1
[1,4,2,5,3] => [1,4,5,2,3] => [2,2,1]
=> 2
[1,4,3,2,5] => [1,4,2,3,5] => [3,1,1]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> 2
[1,4,5,2,3] => [1,3,5,4,2] => [3,1,1]
=> 1
[8,7,5,4,6,3,2,1] => [8,1,2,6,4,3,5,7] => ?
=> ? = 1
[8,5,6,4,7,3,2,1] => [8,1,2,7,6,4,3,5] => ?
=> ? = 1
[8,5,4,6,7,3,2,1] => [8,1,2,6,4,7,3,5] => ?
=> ? = 1
[8,7,5,3,4,6,2,1] => [8,1,4,6,3,2,5,7] => ?
=> ? = 1
[7,8,5,3,4,6,2,1] => [8,1,4,6,3,2,7,5] => ?
=> ? = 1
[8,4,5,6,3,7,2,1] => ? => ?
=> ? = 1
[8,5,6,3,4,7,2,1] => [8,1,4,7,6,3,2,5] => ?
=> ? = 1
[8,4,5,3,6,7,2,1] => ? => ?
=> ? = 1
[8,4,3,5,6,7,2,1] => [8,1,5,3,6,7,2,4] => ?
=> ? = 1
[7,6,4,5,3,8,2,1] => [8,1,7,5,3,4,6,2] => ?
=> ? = 2
[7,5,6,3,4,8,2,1] => ? => ?
=> ? = 2
[7,5,4,3,6,8,2,1] => [8,1,6,3,4,7,5,2] => ?
=> ? = 2
[6,7,8,5,4,2,3,1] => ? => ?
=> ? = 1
[8,6,4,5,7,2,3,1] => [8,3,1,5,7,4,2,6] => ?
=> ? = 1
[7,8,5,6,3,2,4,1] => [8,4,2,1,6,3,7,5] => ?
=> ? = 1
[8,5,6,7,3,2,4,1] => [8,4,2,1,6,7,3,5] => ?
=> ? = 1
[8,7,6,4,3,2,5,1] => [8,5,2,3,1,4,6,7] => ?
=> ? = 1
[8,6,7,4,3,2,5,1] => [8,5,2,3,1,7,4,6] => ?
=> ? = 1
[8,7,4,5,3,2,6,1] => ? => ?
=> ? = 1
[7,8,4,5,3,2,6,1] => ? => ?
=> ? = 1
[8,7,5,3,2,4,6,1] => ? => ?
=> ? = 1
[7,8,5,3,2,4,6,1] => ? => ?
=> ? = 1
[8,7,4,3,2,5,6,1] => [8,5,2,3,6,1,4,7] => ?
=> ? = 1
[8,7,3,2,4,5,6,1] => ? => ?
=> ? = 1
[8,5,6,4,3,2,7,1] => [8,7,2,3,6,4,1,5] => ?
=> ? = 1
[8,5,4,6,3,2,7,1] => [8,7,2,6,4,3,1,5] => ?
=> ? = 1
[8,6,5,3,4,2,7,1] => [8,7,4,2,3,5,1,6] => ?
=> ? = 1
[8,6,3,4,5,2,7,1] => [8,7,4,5,2,3,1,6] => ?
=> ? = 1
[8,6,5,4,2,3,7,1] => [8,3,7,2,4,5,1,6] => ?
=> ? = 1
[8,6,4,5,2,3,7,1] => [8,3,7,5,2,4,1,6] => ?
=> ? = 1
[8,6,5,3,2,4,7,1] => [8,4,2,7,3,5,1,6] => ?
=> ? = 1
[8,6,3,4,2,5,7,1] => [8,5,4,2,7,3,1,6] => ?
=> ? = 1
[8,6,4,2,3,5,7,1] => [8,3,5,2,7,4,1,6] => ?
=> ? = 1
[8,4,5,3,2,6,7,1] => [8,6,2,5,3,7,1,4] => ?
=> ? = 1
[8,4,3,2,5,6,7,1] => [8,5,2,3,6,7,1,4] => ?
=> ? = 1
[6,7,5,4,3,2,8,1] => [8,7,2,3,4,6,5,1] => ?
=> ? = 2
[7,5,6,4,3,2,8,1] => [8,7,2,3,6,4,5,1] => ?
=> ? = 2
[7,6,4,5,3,2,8,1] => [8,7,2,5,3,4,6,1] => ?
=> ? = 2
[7,5,4,6,3,2,8,1] => [8,7,2,6,4,3,5,1] => ?
=> ? = 2
[7,4,5,6,3,2,8,1] => [8,7,2,5,6,3,4,1] => ?
=> ? = 2
[7,6,5,3,4,2,8,1] => [8,7,4,2,3,5,6,1] => ?
=> ? = 2
[6,7,3,4,5,2,8,1] => ? => ?
=> ? = 2
[6,7,4,5,2,3,8,1] => [8,3,7,5,2,6,4,1] => ?
=> ? = 2
[6,7,5,3,2,4,8,1] => [8,4,2,7,3,6,5,1] => ?
=> ? = 2
[6,7,4,3,2,5,8,1] => [8,5,2,3,7,6,4,1] => ?
=> ? = 2
[7,6,3,4,2,5,8,1] => [8,5,4,2,7,3,6,1] => ?
=> ? = 2
[6,7,3,4,2,5,8,1] => ? => ?
=> ? = 2
[6,7,4,2,3,5,8,1] => [8,3,5,2,7,6,4,1] => ?
=> ? = 2
[7,6,2,3,4,5,8,1] => [8,3,4,5,7,2,6,1] => ?
=> ? = 2
[7,5,4,3,2,6,8,1] => [8,6,2,3,4,7,5,1] => ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000834
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 0
[1,2] => [1,0,1,0]
=> [2,1] => [2,1] => 0
[2,1] => [1,1,0,0]
=> [1,2] => [1,2] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [7,6,5,4,2,3,1] => ? = 1
[1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [7,6,5,3,4,2,1] => ? = 1
[1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [7,6,5,3,2,4,1] => ? = 1
[1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => ? = 1
[1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 1
[1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 1
[1,2,3,6,4,5,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 1
[1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [7,6,5,2,3,1,4] => ? = 2
[1,2,3,6,5,4,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 1
[1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [7,6,5,2,3,1,4] => ? = 2
[1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 1
[1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 1
[1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [7,6,4,5,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [7,6,4,5,3,1,2] => ? = 2
[1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [7,6,4,5,2,3,1] => ? = 2
[1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [7,6,4,5,2,1,3] => ? = 2
[1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [7,6,4,3,5,2,1] => ? = 1
[1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [7,6,4,3,5,1,2] => ? = 2
[1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [7,6,4,3,2,5,1] => ? = 1
[1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [7,6,4,3,2,1,5] => ? = 1
[1,2,4,5,7,3,6] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => ? = 1
[1,2,4,5,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [7,6,4,3,1,2,5] => ? = 1
[1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [7,6,4,2,3,5,1] => ? = 1
[1,2,4,6,3,7,5] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [7,6,4,2,3,1,5] => ? = 2
[1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => [7,6,4,2,3,5,1] => ? = 1
[1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [7,6,4,2,3,1,5] => ? = 2
[1,2,4,6,7,3,5] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 1
[1,2,4,6,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 1
[1,2,4,7,3,5,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[1,2,4,7,3,6,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[1,2,4,7,5,3,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[1,2,4,7,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[1,2,4,7,6,3,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 1
[1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 1
[1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [7,6,3,4,5,1,2] => ? = 2
[1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [7,6,3,4,2,5,1] => ? = 2
[1,2,5,3,6,7,4] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => ? = 2
[1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 2
[1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 2
[1,2,5,4,3,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 1
[1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [7,6,3,4,5,1,2] => ? = 2
[1,2,5,4,6,3,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,6,2,1] => [7,6,3,4,2,5,1] => ? = 2
[1,2,5,4,6,7,3] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [7,6,3,4,2,1,5] => ? = 2
[1,2,5,4,7,3,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 2
[1,2,5,4,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 2
[1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [7,6,3,2,4,5,1] => ? = 1
[1,2,5,6,3,7,4] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => ? = 2
[1,2,5,6,4,3,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [7,6,3,2,4,5,1] => ? = 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000251
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> {{1}}
=> {{1}}
=> ? = 0
[1,2] => [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[2,1] => [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 1
[5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[7,8,5,4,6,3,2,1] => ?
=> ?
=> ?
=> ? = 1
[7,8,4,5,6,3,2,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[7,8,4,3,5,6,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,4,5,6,3,7,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,5,6,3,4,7,2,1] => ?
=> ?
=> ?
=> ? = 1
[8,4,5,3,6,7,2,1] => ?
=> ?
=> ?
=> ? = 1
[7,6,5,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[7,6,4,5,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[7,6,5,3,4,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[7,5,6,3,4,8,2,1] => ?
=> ?
=> ?
=> ? = 2
[7,6,4,3,5,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[7,6,3,4,5,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[7,5,4,3,6,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[7,5,3,4,6,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 2
[3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 1
[6,7,8,5,4,2,3,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,7,8,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[7,8,5,6,3,2,4,1] => ?
=> ?
=> ?
=> ? = 1
[5,6,7,8,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[5,6,7,8,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[8,7,6,4,2,3,5,1] => ?
=> ?
=> ?
=> ? = 1
[8,6,7,4,2,3,5,1] => ?
=> ?
=> ?
=> ? = 1
[8,7,4,5,3,2,6,1] => ?
=> ?
=> ?
=> ? = 1
[7,8,4,5,3,2,6,1] => ?
=> ?
=> ?
=> ? = 1
[7,8,5,3,2,4,6,1] => ?
=> ?
=> ?
=> ? = 1
[8,7,3,2,4,5,6,1] => ?
=> ?
=> ?
=> ? = 1
[8,6,4,5,2,3,7,1] => ?
=> ?
=> ?
=> ? = 1
[8,6,3,4,2,5,7,1] => ?
=> ?
=> ?
=> ? = 1
[6,7,5,4,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 2
[7,5,4,6,3,2,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,3,4,5,2,8,1] => ?
=> ?
=> ?
=> ? = 2
[5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[6,7,4,5,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 2
[6,7,5,3,2,4,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 2
[7,6,4,3,2,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,4,3,2,5,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 2
[7,6,3,4,2,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,3,4,2,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,4,2,3,5,8,1] => ?
=> ?
=> ?
=> ? = 2
[6,7,3,2,4,5,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 2
[6,7,2,3,4,5,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> {{1,8},{2,4,5,6,7},{3}}
=> ? = 2
[7,4,5,3,2,6,8,1] => ?
=> ?
=> ?
=> ? = 2
[2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[5,6,7,8,4,3,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[7,8,5,4,6,3,1,2] => ?
=> ?
=> ?
=> ? = 1
[5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 1
[8,7,6,4,3,5,1,2] => ?
=> ?
=> ?
=> ? = 1
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000994
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [[1]]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[2,5,1,3,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,1,4,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,3,1,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,3,4,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,4,1,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,4,3,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,6,1,3,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,3,5,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,4,3,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,4,5,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,5,3,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,5,4,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,1,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,1,5,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,4,1,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,4,5,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,5,1,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,5,4,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,1,3,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,1,5,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,3,1,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,3,5,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,5,1,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,5,3,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,1,3,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,1,4,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,3,1,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,3,4,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,4,1,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,4,3,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,2,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,2,5,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,4,2,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,4,5,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,5,2,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,5,4,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,1,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,1,5,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,4,1,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,4,5,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,5,1,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,5,4,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,1,2,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,1,5,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,2,1,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,2,5,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,5,1,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,5,2,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,5,1,2,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,5,1,4,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [[1]]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => 1
[2,5,1,3,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,1,4,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,3,1,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,3,4,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,4,1,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,5,4,3,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,2,3,4,6,7] => ? = 1
[2,6,1,3,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,3,5,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,4,3,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,4,5,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,5,3,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,1,5,4,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,1,4,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,1,5,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,4,1,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,4,5,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,5,1,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,3,5,4,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,1,3,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,1,5,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,3,1,5,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,3,5,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,5,1,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,4,5,3,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,1,3,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,1,4,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,3,1,4,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,3,4,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,4,1,3,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[2,6,5,4,3,1,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,2,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,2,5,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,4,2,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,4,5,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,5,2,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,1,5,4,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,1,4,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,1,5,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,4,1,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,4,5,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,5,1,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,2,5,4,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,1,2,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,1,5,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,2,1,5,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,2,5,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,5,1,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,4,5,2,1,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,5,1,2,4,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
[3,6,5,1,4,2,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7] => ? = 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [[1]]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,5,2,3,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,2,4,3,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,3,2,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,4,2,3,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,3,4,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,3,5,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,4,3,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,4,5,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,5,3,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,5,4,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,2,4,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,2,5,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,4,2,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,4,5,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,5,2,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,5,4,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,2,3,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,2,5,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,3,2,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,3,5,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,5,2,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,5,3,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,2,3,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,2,4,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,3,2,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,3,4,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,4,2,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,4,3,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[2,1,3,4,7,5,6] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [2,1,3,4,7,6,5] => ? = 2
[2,1,3,4,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [2,1,3,4,7,6,5] => ? = 2
[2,1,3,7,4,5,6] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,4,6,5] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,5,4,6] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,5,6,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,6,4,5] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,6,5,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,6,3,4,5,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,3,5,4,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,4,3,5,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,4,5,3,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,5,3,4,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,5,4,3,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,5,1,3,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,1,4,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,3,1,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,3,4,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,4,1,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,4,3,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000670
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [[1]]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 1
[1,5,2,3,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,2,4,3,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,3,2,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,4,2,3,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,5,4,3,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,3,4,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,3,5,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,4,3,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,4,5,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,5,3,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,2,5,4,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,2,4,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,2,5,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,4,2,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,4,5,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,5,2,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,3,5,4,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,2,3,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,2,5,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,3,2,7,5] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,3,5,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,5,2,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,4,5,3,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,2,3,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,2,4,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,3,2,7,4] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,3,4,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,4,2,7,3] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[1,6,5,4,3,7,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,5,4,3,2,7,6] => ? = 2
[2,1,3,4,7,5,6] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [2,1,3,4,7,6,5] => ? = 2
[2,1,3,4,7,6,5] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [2,1,3,4,7,6,5] => ? = 2
[2,1,3,7,4,5,6] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,4,6,5] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,5,4,6] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,5,6,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,6,4,5] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,3,7,6,5,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [2,1,3,7,6,5,4] => ? = 2
[2,1,6,3,4,5,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,3,5,4,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,4,3,5,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,4,5,3,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,5,3,4,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,1,6,5,4,3,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> [2,1,6,5,4,3,7] => ? = 2
[2,5,1,3,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,1,4,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,3,1,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,3,4,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,4,1,3,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
[2,5,4,3,1,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,5,4,3,2,6,7] => ? = 1
Description
The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000386The number of factors DDU in a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St000702The number of weak deficiencies of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000155The number of exceedances (also excedences) of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001874Lusztig's a-function for the symmetric group. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001720The minimal length of a chain of small intervals in a lattice. St000092The number of outer peaks of a permutation. St000353The number of inner valleys of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001935The number of ascents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice.
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