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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
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(load all 12 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,2] => [1,1]
=> 2
[2,1] => [2]
=> 0
[1,2,3] => [1,1,1]
=> 3
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 0
[3,1,2] => [3]
=> 0
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3] => [2,1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> 2
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> 0
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [4]
=> 0
[2,4,1,3] => [4]
=> 0
[2,4,3,1] => [3,1]
=> 1
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [4]
=> 0
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [3,1]
=> 1
[3,4,1,2] => [2,2]
=> 0
[3,4,2,1] => [4]
=> 0
[4,1,2,3] => [4]
=> 0
[4,1,3,2] => [3,1]
=> 1
[4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [4]
=> 0
[4,3,2,1] => [2,2]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [2,1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> 3
[1,2,4,5,3] => [3,1,1]
=> 2
[1,2,5,3,4] => [3,1,1]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> 2
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [4,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [3,1,1]
=> 2
[1,4,2,5,3] => [4,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> 3
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [2,2,1]
=> 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000674
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 79%●distinct values known / distinct values provided: 69%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 79%●distinct values known / distinct values provided: 69%
Values
[1] => [1]
=> [1]
=> [1,0]
=> ? = 1
[1,2] => [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[2,1] => [2]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,2] => [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,2,1] => [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,4,2] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,4,3] => [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,3,1,4] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,3,1] => [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,3,2,1] => [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,2,5,3] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[7,6,5,8,4,3,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,5,7,8,4,3,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,6,4,5,8,3,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,4,5,6,8,3,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,5,4,7,8,3,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,6,8,5,3,4,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,3,4,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,6,8,3,4,5,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,5,4,6,3,8,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,7,3,8,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,3,4,5,6,8,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,3,4,6,7,8,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,4,5,6,7,8,2,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,5,6,8,4,2,3,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,4,5,6,2,3,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,7,8,2,3,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,5,3,2,4,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,5,3,2,4,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,6,8,5,2,3,4,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,5,6,2,3,4,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,5,8,2,3,4,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,2,3,4,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,3,4,2,5,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,3,4,2,5,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,6,2,3,4,5,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,8,2,3,4,5,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,4,2,3,5,6,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[7,8,2,3,4,5,6,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,5,4,7,3,2,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,5,7,3,4,2,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,4,5,3,7,2,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,3,7,2,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,3,4,5,7,2,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,3,5,6,7,2,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,4,5,2,3,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,4,7,2,3,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,5,3,2,4,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,2,3,4,5,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,4,6,3,2,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,4,5,6,2,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,6,2,3,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,3,5,2,6,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[4,5,2,3,6,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[3,4,2,5,6,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[2,3,4,5,6,7,8,1] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,5,7,4,3,1,2] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[6,7,5,8,4,3,1,2] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[5,6,7,8,4,3,1,2] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
[8,6,4,5,7,3,1,2] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000445
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 85%
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 85%
Values
[1] => {{1}}
=> [1] => [1,0]
=> 1
[1,2] => {{1},{2}}
=> [1,1] => [1,0,1,0]
=> 2
[2,1] => {{1,2}}
=> [2] => [1,1,0,0]
=> 0
[1,2,3] => {{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,2] => {{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => {{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => {{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
[3,2,1] => {{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,3,1] => {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[3,2,1,4] => {{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,2,3] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => {{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,1,2] => {{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[4,3,2,1] => {{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> ? => ?
=> ? = 0
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 0
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 0
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> ? => ?
=> ? = 2
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 0
[6,7,5,4,8,3,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[7,5,6,4,8,3,2,1] => {{1,2,5,7,8},{3,6},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ? => ?
=> ? = 1
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ? => ?
=> ? = 0
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 0
[7,8,6,4,3,5,2,1] => {{1,2,7,8},{3,5,6},{4}}
=> ? => ?
=> ? = 1
[6,7,8,4,3,5,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[8,7,6,3,4,5,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> ? => ?
=> ? = 0
[8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
=> ? => ?
=> ? = 2
[7,8,5,3,4,6,2,1] => {{1,2,7,8},{3,4,5},{6}}
=> ? => ?
=> ? = 1
[8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ?
=> ? = 0
[7,6,5,4,3,8,2,1] => {{1,2,6,7,8},{3,5},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[6,5,7,4,3,8,2,1] => {{1,6,8},{2,3,5,7},{4}}
=> ? => ?
=> ? = 1
[6,7,4,5,3,8,2,1] => {{1,6,8},{2,7},{3,4,5}}
=> ? => ?
=> ? = 0
[6,4,5,7,3,8,2,1] => {{1,6,8},{2,4,7},{3,5}}
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[7,6,4,3,5,8,2,1] => {{1,2,6,7,8},{3,4},{5}}
=> ? => ?
=> ? = 1
[6,5,4,3,7,8,2,1] => {{1,6,8},{2,5,7},{3,4}}
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[7,6,8,5,4,2,3,1] => {{1,3,7,8},{2,6},{4,5}}
=> ? => ?
=> ? = 0
[8,6,5,7,4,2,3,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ?
=> ? = 0
[8,6,7,4,5,2,3,1] => {{1,8},{2,6},{3,7},{4},{5}}
=> ? => ?
=> ? = 2
[8,6,4,5,7,2,3,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ?
=> ? = 0
[8,4,5,6,7,2,3,1] => {{1,8},{2,4,6},{3,5,7}}
=> ? => ?
=> ? = 0
[7,6,5,4,8,2,3,1] => {{1,3,5,7,8},{2,6},{4}}
=> ? => ?
=> ? = 1
[6,5,7,4,8,2,3,1] => {{1,2,5,6,8},{3,7},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[8,6,7,5,3,2,4,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ?
=> ? = 0
[8,7,5,6,3,2,4,1] => {{1,8},{2,4,6,7},{3,5}}
=> ? => ?
=> ? = 0
[8,5,6,7,3,2,4,1] => {{1,8},{2,3,5,6},{4,7}}
=> ? => ?
=> ? = 0
[8,7,6,5,2,3,4,1] => {{1,8},{2,4,5,7},{3,6}}
=> ? => ?
=> ? = 0
[8,6,5,7,2,3,4,1] => {{1,8},{2,3,5,6},{4,7}}
=> ? => ?
=> ? = 0
[7,6,8,4,3,2,5,1] => {{1,3,5,7,8},{2,6},{4}}
=> ? => ?
=> ? = 1
[8,6,7,3,4,2,5,1] => {{1,8},{2,6},{3,4,5,7}}
=> ? => ?
=> ? = 0
[6,7,8,4,2,3,5,1] => {{1,3,6,8},{2,5,7},{4}}
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[8,7,6,3,2,4,5,1] => {{1,8},{2,5,7},{3,4,6}}
=> ? => ?
=> ? = 0
[8,6,7,2,3,4,5,1] => {{1,8},{2,4,6},{3,5,7}}
=> ? => ?
=> ? = 0
[8,7,5,3,4,2,6,1] => {{1,8},{2,6,7},{3,4,5}}
=> ? => ?
=> ? = 0
[7,8,4,3,5,2,6,1] => {{1,2,6,7,8},{3,4},{5}}
=> ? => ?
=> ? = 1
[8,7,4,3,2,5,6,1] => {{1,8},{2,5,6,7},{3,4}}
=> ? => ?
=> ? = 0
[8,6,5,4,3,2,7,1] => {{1,8},{2,6},{3,5},{4},{7}}
=> ? => ?
=> ? = 2
[8,6,3,4,5,2,7,1] => {{1,8},{2,6},{3},{4},{5},{7}}
=> ? => ?
=> ? = 4
[8,5,4,3,2,6,7,1] => {{1,8},{2,5},{3,4},{6},{7}}
=> ? => ?
=> ? = 2
[8,3,4,5,2,6,7,1] => {{1,8},{2,3,4,5},{6},{7}}
=> ? => ?
=> ? = 2
[8,5,4,2,3,6,7,1] => {{1,8},{2,3,4,5},{6},{7}}
=> ? => ?
=> ? = 2
[8,4,5,2,3,6,7,1] => {{1,8},{2,4},{3,5},{6},{7}}
=> ? => ?
=> ? = 2
[6,7,5,4,3,2,8,1] => {{1,2,6,7,8},{3,5},{4}}
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[7,5,6,4,3,2,8,1] => {{1,7,8},{2,3,5,6},{4}}
=> ? => ?
=> ? = 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000247
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 62%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 62%
Values
[1] => [1]
=> [[1]]
=> {{1}}
=> ? = 1
[1,2] => [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[2,1] => [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,1,2] => [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,3,4,1] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[2,4,1,3] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,1,4,2] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[3,4,2,1] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,1,2,3] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[4,3,1,2] => [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[8,7,6,5,4,3,2,1] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[7,8,6,5,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[8,6,7,5,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[6,7,8,5,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[8,7,5,6,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[7,8,5,6,4,3,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[8,5,6,7,4,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[5,6,7,8,4,3,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[8,7,6,4,5,3,2,1] => [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 2
[7,8,6,4,5,3,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,6,7,4,5,3,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,7,5,4,6,3,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[8,7,4,5,6,3,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[8,6,5,4,7,3,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[8,5,6,4,7,3,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[6,7,5,4,8,3,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[7,5,6,4,8,3,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[5,6,7,4,8,3,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 1
[7,5,4,6,8,3,2,1] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 0
[5,6,4,7,8,3,2,1] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 0
[6,4,5,7,8,3,2,1] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 0
[5,4,6,7,8,3,2,1] => [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 0
[8,7,6,5,3,4,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[7,8,6,5,3,4,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[8,7,5,6,3,4,2,1] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[7,8,5,6,3,4,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[6,5,7,8,3,4,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[8,7,6,4,3,5,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[7,8,6,4,3,5,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 1
[8,6,7,4,3,5,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[6,7,8,4,3,5,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[8,7,6,3,4,5,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[7,8,6,3,4,5,2,1] => [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[8,7,5,4,3,6,2,1] => [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 2
[7,8,5,4,3,6,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,7,4,5,3,6,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[7,8,5,3,4,6,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 1
[8,7,4,3,5,6,2,1] => [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 2
[8,7,3,4,5,6,2,1] => [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 4
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 4
[8,6,5,4,3,7,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[8,6,4,3,5,7,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[8,5,4,3,6,7,2,1] => [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 0
[8,5,3,4,6,7,2,1] => [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 2
[8,4,3,5,6,7,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[7,6,5,4,3,8,2,1] => [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[6,7,5,4,3,8,2,1] => [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[6,5,7,4,3,8,2,1] => [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 1
[6,7,4,5,3,8,2,1] => [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 0
Description
The number of singleton blocks of a set partition.
Matching statistic: St000241
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 54%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 54%
Values
[1] => [1]
=> [1,0]
=> [1] => 1
[1,2] => [1,1]
=> [1,1,0,0]
=> [2,1] => 2
[2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 7
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,5,6,7,4] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,3,5,7,4,6] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,6,4,7,5] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,6,7,5,4] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,3,7,4,5,6] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,3,7,6,4,5] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,4,5,6,3,7] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,5,6,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,4,5,7,3,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,4,5,7,6,3] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,6,3,5,7] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,6,3,7,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,4,6,5,7,3] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,6,7,5,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,4,7,3,5,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,4,7,3,6,5] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,7,5,3,6] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,4,7,6,3,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,5,3,6,4,7] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,5,3,6,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,5,3,7,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,5,3,7,6,4] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,5,4,3,6,7] => [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 5
[1,2,5,4,6,3,7] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,5,4,6,7,3] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,5,4,7,3,6] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,5,4,7,6,3] => [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? = 4
[1,2,5,6,4,3,7] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? = 3
[1,2,5,6,4,7,3] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
[1,2,5,6,7,3,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? = 2
Description
The number of cyclical small excedances.
A cyclical small excedance is an index i such that πi=i+1 considered cyclically.
Matching statistic: St000022
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 85%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 85%
Values
[1] => {{1}}
=> [1] => [1] => 1
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 2
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 0
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 3
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
[3,1,2] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 0
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 0
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 4
[1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 4
[1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 5
[1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 5
[1,2,3,5,4,7,6] => {{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 3
[1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 4
[1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,2,3,5,7,4,6] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,2,3,5,7,6,4] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 4
[1,2,3,6,4,5,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 4
[1,2,3,6,4,7,5] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 5
[1,2,3,6,5,7,4] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 4
[1,2,3,6,7,4,5] => {{1},{2},{3},{4,6},{5,7}}
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ? = 3
[1,2,3,6,7,5,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,2,3,7,4,5,6] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,2,3,7,4,6,5] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 4
[1,2,3,7,5,4,6] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 4
[1,2,3,7,5,6,4] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 5
[1,2,3,7,6,4,5] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,2,3,7,6,5,4] => {{1},{2},{3},{4,7},{5,6}}
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 5
[1,2,4,3,5,7,6] => {{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 3
[1,2,4,3,6,5,7] => {{1},{2},{3,4},{5,6},{7}}
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 3
[1,2,4,3,6,7,5] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 2
[1,2,4,3,7,5,6] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 2
[1,2,4,3,7,6,5] => {{1},{2},{3,4},{5,7},{6}}
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3
[1,2,4,5,3,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 4
[1,2,4,5,3,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 2
[1,2,4,5,6,3,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 3
[1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,4,5,7,3,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,4,5,7,6,3] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 3
[1,2,4,6,3,5,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 3
[1,2,4,6,3,7,5] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,4,6,5,3,7] => {{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => ? = 4
[1,2,4,6,5,7,3] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 3
[1,2,4,6,7,3,5] => {{1},{2},{3,4,6},{5,7}}
=> [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => ? = 2
[1,2,4,6,7,5,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,4,7,3,5,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,4,7,3,6,5] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 3
[1,2,4,7,5,3,6] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 3
[1,2,4,7,5,6,3] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => ? = 4
[1,2,4,7,6,3,5] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,4,7,6,5,3] => {{1},{2},{3,4,7},{5,6}}
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => ? = 2
[1,2,5,3,4,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 4
[1,2,5,3,4,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 2
[1,2,5,3,6,4,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 3
[1,2,5,3,6,7,4] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
[1,2,5,3,7,4,6] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 2
Description
The number of fixed points of a permutation.
Matching statistic: St000248
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 69%
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 69%
Values
[1] => {{1}}
=> {{1}}
=> {{1}}
=> ? = 1
[1,2] => {{1},{2}}
=> {{1,2}}
=> {{1,2}}
=> 2
[2,1] => {{1,2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
[1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 3
[1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[3,1,2] => {{1,2,3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[2,4,3,1] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[3,2,1,4] => {{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[3,4,2,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[4,1,3,2] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 2
[4,3,1,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[4,3,2,1] => {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 3
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1},{2,8},{3,7},{4,6},{5}}
=> {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 0
[7,8,6,5,4,3,2,1] => {{1,2,7,8},{3,6},{4,5}}
=> {{1},{2},{3,7},{4,6},{5},{8}}
=> {{1},{2,6},{3,5},{4},{7},{8}}
=> ? = 0
[8,6,7,5,4,3,2,1] => {{1,8},{2,3,6,7},{4,5}}
=> {{1},{2,8},{3},{4,6},{5},{7}}
=> ?
=> ? = 0
[7,6,8,5,4,3,2,1] => {{1,2,3,6,7,8},{4,5}}
=> {{1},{2},{3},{4,6},{5},{7},{8}}
=> ?
=> ? = 0
[6,7,8,5,4,3,2,1] => {{1,3,6,8},{2,7},{4,5}}
=> {{1},{2,7},{3,8},{4,6},{5}}
=> {{1,6},{2,7},{3,5},{4},{8}}
=> ? = 0
[8,7,5,6,4,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> {{1},{2,8},{3,7},{4},{5},{6}}
=> {{1,7},{2,6},{3},{4},{5},{8}}
=> ? = 0
[7,8,5,6,4,3,2,1] => {{1,2,7,8},{3,4,5,6}}
=> {{1},{2},{3,7},{4},{5},{6},{8}}
=> ?
=> ? = 0
[8,6,5,7,4,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[8,5,6,7,4,3,2,1] => {{1,8},{2,4,5,7},{3,6}}
=> {{1},{2,8},{3,6},{4,7},{5}}
=> {{1,7},{2,5},{3,6},{4},{8}}
=> ? = 0
[7,6,5,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,5,8,4,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> {{1},{2,7},{3,8},{4},{5},{6}}
=> {{1,6},{2,7},{3},{4},{5},{8}}
=> ? = 0
[7,5,6,8,4,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> {{1},{2},{3,6},{4,7},{5},{8}}
=> ?
=> ? = 0
[6,5,7,8,4,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[5,6,7,8,4,3,2,1] => {{1,4,5,8},{2,3,6,7}}
=> {{1},{2,6},{3},{4,8},{5},{7}}
=> {{1,5},{2},{3,7},{4},{6},{8}}
=> ? = 0
[8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
=> {{1},{2,8},{3,7},{4,5,6}}
=> {{1,7},{2,6},{3,4,5},{8}}
=> ? = 2
[7,8,6,4,5,3,2,1] => {{1,2,7,8},{3,6},{4},{5}}
=> {{1},{2},{3,7},{4,5,6},{8}}
=> ?
=> ? = 2
[8,6,7,4,5,3,2,1] => {{1,8},{2,3,6,7},{4},{5}}
=> {{1},{2,8},{3},{4,5,6},{7}}
=> {{1,7},{2},{3,4,5},{6},{8}}
=> ? = 2
[7,6,8,4,5,3,2,1] => {{1,2,3,6,7,8},{4},{5}}
=> {{1},{2},{3},{4,5,6},{7},{8}}
=> {{1},{2},{3,4,5},{6},{7},{8}}
=> ? = 2
[8,7,5,4,6,3,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> {{1},{2,8},{3,7},{4},{5,6}}
=> {{1,7},{2,6},{3,4},{5},{8}}
=> ? = 1
[8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> {{1},{2,8},{3,7},{4},{5},{6}}
=> {{1,7},{2,6},{3},{4},{5},{8}}
=> ? = 0
[8,6,5,4,7,3,2,1] => {{1,8},{2,3,5,6,7},{4}}
=> {{1},{2,8},{3},{4},{5,6},{7}}
=> {{1,7},{2},{3,4},{5},{6},{8}}
=> ? = 1
[8,5,6,4,7,3,2,1] => {{1,8},{2,5,7},{3,6},{4}}
=> {{1},{2,8},{3,5,6},{4,7}}
=> {{1,7},{2,5},{3,4,6},{8}}
=> ? = 1
[8,6,4,5,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[8,4,5,6,7,3,2,1] => {{1,8},{2,3,4,5,6,7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[7,6,5,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}}
=> {{1},{2},{3},{4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 1
[6,7,5,4,8,3,2,1] => {{1,3,5,6,8},{2,7},{4}}
=> {{1},{2,7},{3,8},{4},{5,6}}
=> {{1,6},{2,7},{3,4},{5},{8}}
=> ? = 1
[7,5,6,4,8,3,2,1] => {{1,2,5,7,8},{3,6},{4}}
=> {{1},{2},{3,5,6},{4,7},{8}}
=> {{1},{2,5},{3,4,6},{7},{8}}
=> ? = 1
[6,5,7,4,8,3,2,1] => {{1,2,3,5,6,7,8},{4}}
=> {{1},{2},{3},{4},{5,6},{7},{8}}
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 1
[5,6,7,4,8,3,2,1] => {{1,5,8},{2,3,6,7},{4}}
=> ?
=> ?
=> ? = 1
[7,6,4,5,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,7,4,5,8,3,2,1] => {{1,3,4,5,6,8},{2,7}}
=> {{1},{2,7},{3,8},{4},{5},{6}}
=> {{1,6},{2,7},{3},{4},{5},{8}}
=> ? = 0
[7,5,4,6,8,3,2,1] => {{1,2,5,7,8},{3,4,6}}
=> {{1},{2},{3,5},{4,7},{6},{8}}
=> ?
=> ? = 0
[7,4,5,6,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[6,5,4,7,8,3,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[5,6,4,7,8,3,2,1] => {{1,5,8},{2,3,4,6,7}}
=> {{1},{2,5},{3},{4,8},{6},{7}}
=> {{1,5},{2},{3},{4,7},{6},{8}}
=> ? = 0
[6,4,5,7,8,3,2,1] => {{1,3,5,6,8},{2,4,7}}
=> {{1},{2,5,7},{3,6,8},{4}}
=> {{1,3,6},{2,4,7},{5},{8}}
=> ? = 0
[5,4,6,7,8,3,2,1] => {{1,5,8},{2,4,7},{3,6}}
=> ?
=> ?
=> ? = 0
[4,5,6,7,8,3,2,1] => {{1,2,4,5,7,8},{3,6}}
=> {{1},{2},{3,6},{4,7},{5},{8}}
=> ?
=> ? = 0
[8,7,6,5,3,4,2,1] => {{1,8},{2,7},{3,4,5,6}}
=> {{1},{2,8},{3,7},{4},{5},{6}}
=> {{1,7},{2,6},{3},{4},{5},{8}}
=> ? = 0
[7,8,6,5,3,4,2,1] => {{1,2,7,8},{3,4,5,6}}
=> {{1},{2},{3,7},{4},{5},{6},{8}}
=> ?
=> ? = 0
[8,6,7,5,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[7,6,8,5,3,4,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,8},{3,5,7},{4,6}}
=> {{1,7},{2,4,6},{3,5},{8}}
=> ? = 0
[7,8,5,6,3,4,2,1] => {{1,2,7,8},{3,5},{4,6}}
=> {{1},{2},{3,5,7},{4,6},{8}}
=> {{1},{2,4,6},{3,5},{7},{8}}
=> ? = 0
[8,5,6,7,3,4,2,1] => {{1,8},{2,3,4,5,6,7}}
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 0
[7,6,5,8,3,4,2,1] => {{1,2,4,6,7,8},{3,5}}
=> {{1},{2},{3},{4,6},{5,7},{8}}
=> ?
=> ? = 0
[6,5,7,8,3,4,2,1] => {{1,4,6,8},{2,3,5,7}}
=> {{1},{2,4,6},{3,5,8},{7}}
=> {{1,4,6},{2},{3,5,7},{8}}
=> ? = 0
[5,6,7,8,3,4,2,1] => {{1,2,3,4,5,6,7,8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[8,7,6,4,3,5,2,1] => {{1,8},{2,7},{3,5,6},{4}}
=> {{1},{2,8},{3,7},{4},{5,6}}
=> {{1,7},{2,6},{3,4},{5},{8}}
=> ? = 1
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition S is an index i such that i and i+1 (considered cyclically) are both in the same block of S.
For noncrossing set partitions, this is also the number of singletons of the image of S under the Kreweras complement.
Matching statistic: St000895
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 54%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 54%
Values
[1] => {{1}}
=> [1] => [[1]]
=> 1
[1,2] => {{1},{2}}
=> [1,2] => [[1,0],[0,1]]
=> 2
[2,1] => {{1,2}}
=> [2,1] => [[0,1],[1,0]]
=> 0
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[3,1,2] => {{1,2,3}}
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [[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]]
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [[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]]
=> 3
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [[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]]
=> 3
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [[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]]
=> 2
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [[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]]
=> 2
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [[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]]
=> 3
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [[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]]
=> 3
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [[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
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [[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]]
=> 2
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [[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]]
=> 2
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [[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]]
=> 3
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 2
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 1
[1,4,6,2,5,3] => {{1},{2,4},{3,6},{5}}
=> [1,4,6,2,5,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,6,5,2,3] => {{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,3,6,2,4] => {{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 2
[1,5,4,6,2,3] => {{1},{2,5},{3,4,6}}
=> [1,5,4,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,5,6,2,4,3] => {{1},{2,4,5},{3,6}}
=> [1,4,6,5,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,6,3,2,4] => {{1},{2,5},{3,4,6}}
=> [1,5,4,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,5,6,4,2,3] => {{1},{2,5},{3,6},{4}}
=> [1,5,6,4,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 2
[2,4,6,1,5,3] => {{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[2,4,6,5,1,3] => {{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[2,5,3,6,1,4] => {{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[2,5,4,6,1,3] => {{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[2,5,6,1,4,3] => {{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[2,5,6,3,1,4] => {{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[2,5,6,4,1,3] => {{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
[3,2,5,6,1,4] => {{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[3,4,1,6,5,2] => {{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[3,4,6,2,5,1] => {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[3,5,1,4,6,2] => {{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[3,5,4,1,6,2] => {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[3,6,1,2,5,4] => {{1,3},{2,4,6},{5}}
=> [3,4,1,6,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[3,6,1,4,2,5] => {{1,3},{2,5,6},{4}}
=> [3,5,1,4,6,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[3,6,1,4,5,2] => {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
[3,6,4,1,2,5] => {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[3,6,4,1,5,2] => {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[3,6,4,5,1,2] => {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[3,6,5,1,4,2] => {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[4,1,6,2,5,3] => {{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[4,1,6,5,2,3] => {{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[4,2,5,1,6,3] => {{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[4,2,5,6,3,1] => {{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[4,2,6,1,3,5] => {{1,4},{2},{3,5,6}}
=> [4,2,5,1,6,3] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[4,2,6,1,5,3] => {{1,4},{2},{3,6},{5}}
=> [4,2,6,1,5,3] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[4,2,6,5,1,3] => {{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[4,3,5,1,6,2] => {{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,3,6,1,2,5] => {{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,3,6,1,5,2] => {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[4,3,6,5,1,2] => {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[4,5,1,3,6,2] => {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,5,2,1,6,3] => {{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,5,3,1,6,2] => {{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[4,5,6,1,3,2] => {{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,5,6,2,1,3] => {{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[4,6,1,3,2,5] => {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,6,1,3,5,2] => {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[4,6,1,5,3,2] => {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[4,6,2,1,3,5] => {{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[4,6,2,1,5,3] => {{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 1
[4,6,2,5,1,3] => {{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[4,6,3,1,2,5] => {{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[4,6,3,1,5,2] => {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000215
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 85%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 85%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [2,1] => [2,1] => 2
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [2,3,1] => [3,1,2] => [3,2,1] => 3
[1,3,2] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 4
[1,2,4,3] => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
[1,3,2,4] => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1
[1,4,2,3] => [3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 1
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[3,1,4,2] => [4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 0
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
[4,1,3,2] => [4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 1
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[4,2,3,1] => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
[4,3,1,2] => [4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 0
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 3
[1,2,4,3,5] => [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 3
[1,2,4,5,3] => [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 2
[1,2,5,3,4] => [2,4,5,3,1] => [5,1,4,2,3] => [5,3,4,2,1] => 2
[1,2,5,4,3] => [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 3
[1,3,2,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 3
[1,3,2,5,4] => [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 1
[1,3,4,2,5] => [4,2,3,5,1] => [5,2,3,1,4] => [2,3,5,4,1] => 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [2,5,3,4,1] => 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,5,1] => 2
[1,4,2,3,5] => [3,4,2,5,1] => [5,3,1,2,4] => [5,4,2,3,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [4,5,2,3,1] => 1
[1,4,3,2,5] => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 3
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,5,1] => 2
[1,4,5,2,3] => [4,5,2,3,1] => [5,3,4,1,2] => [5,2,3,4,1] => 1
[1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => [6,7,5,4,3,2,1] => ? = 5
[1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => [5,7,6,4,3,2,1] => ? = 5
[1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [5,6,7,4,3,2,1] => ? = 4
[1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => [7,5,6,4,3,2,1] => ? = 4
[1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [6,5,7,4,3,2,1] => ? = 5
[1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => [4,7,6,5,3,2,1] => ? = 5
[1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => [4,6,7,5,3,2,1] => ? = 3
[1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => [4,5,7,6,3,2,1] => ? = 4
[1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => [4,5,6,7,3,2,1] => ? = 3
[1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => [7,1,2,4,6,3,5] => [4,7,5,6,3,2,1] => ? = 3
[1,2,3,5,7,6,4] => [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => [4,6,5,7,3,2,1] => ? = 4
[1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => [7,1,2,5,3,4,6] => [7,6,4,5,3,2,1] => ? = 4
[1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => [7,1,2,5,3,6,4] => [6,7,4,5,3,2,1] => ? = 3
[1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => [5,4,7,6,3,2,1] => ? = 5
[1,2,3,6,5,7,4] => [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => [5,4,6,7,3,2,1] => ? = 4
[1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => [7,1,2,5,6,3,4] => [7,4,5,6,3,2,1] => ? = 3
[1,2,3,6,7,5,4] => [2,3,7,6,4,5,1] => [7,1,2,5,6,4,3] => [6,4,5,7,3,2,1] => ? = 3
[1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => ? = 3
[1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => [7,4,6,5,3,2,1] => ? = 4
[1,2,3,7,5,4,6] => [2,3,6,5,7,4,1] => [7,1,2,6,4,3,5] => [7,5,4,6,3,2,1] => ? = 4
[1,2,3,7,5,6,4] => [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => [6,5,4,7,3,2,1] => ? = 5
[1,2,3,7,6,4,5] => [2,3,6,7,5,4,1] => [7,1,2,6,5,3,4] => [5,7,4,6,3,2,1] => ? = 3
[1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [5,6,4,7,3,2,1] => ? = 3
[1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => [3,7,6,5,4,2,1] => ? = 5
[1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => [3,6,7,5,4,2,1] => ? = 3
[1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => [7,1,3,2,5,4,6] => [3,5,7,6,4,2,1] => ? = 3
[1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => [7,1,3,2,5,6,4] => [3,5,6,7,4,2,1] => ? = 2
[1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => [7,1,3,2,6,4,5] => [3,7,5,6,4,2,1] => ? = 2
[1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => [7,1,3,2,6,5,4] => [3,6,5,7,4,2,1] => ? = 3
[1,2,4,5,3,6,7] => [2,5,3,4,6,7,1] => [7,1,3,4,2,5,6] => [3,4,7,6,5,2,1] => ? = 4
[1,2,4,5,3,7,6] => [2,5,3,4,7,6,1] => [7,1,3,4,2,6,5] => [3,4,6,7,5,2,1] => ? = 2
[1,2,4,5,6,3,7] => [2,6,3,4,5,7,1] => [7,1,3,4,5,2,6] => [3,4,5,7,6,2,1] => ? = 3
[1,2,4,5,6,7,3] => [2,7,3,4,5,6,1] => [7,1,3,4,5,6,2] => [3,4,5,6,7,2,1] => ? = 2
[1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => [7,1,3,4,6,2,5] => [3,4,7,5,6,2,1] => ? = 2
[1,2,4,5,7,6,3] => [2,7,3,4,6,5,1] => [7,1,3,4,6,5,2] => [3,4,6,5,7,2,1] => ? = 3
[1,2,4,6,3,5,7] => [2,5,3,6,4,7,1] => [7,1,3,5,2,4,6] => [3,7,6,4,5,2,1] => ? = 3
[1,2,4,6,3,7,5] => [2,5,3,7,4,6,1] => [7,1,3,5,2,6,4] => [3,6,7,4,5,2,1] => ? = 2
[1,2,4,6,5,3,7] => [2,6,3,5,4,7,1] => [7,1,3,5,4,2,6] => [3,5,4,7,6,2,1] => ? = 4
[1,2,4,6,5,7,3] => [2,7,3,5,4,6,1] => [7,1,3,5,4,6,2] => [3,5,4,6,7,2,1] => ? = 3
[1,2,4,6,7,3,5] => [2,6,3,7,4,5,1] => [7,1,3,5,6,2,4] => [3,7,4,5,6,2,1] => ? = 2
[1,2,4,6,7,5,3] => [2,7,3,6,4,5,1] => [7,1,3,5,6,4,2] => [3,6,4,5,7,2,1] => ? = 2
[1,2,4,7,3,5,6] => [2,5,3,6,7,4,1] => [7,1,3,6,2,4,5] => [3,6,4,7,5,2,1] => ? = 2
[1,2,4,7,3,6,5] => [2,5,3,7,6,4,1] => [7,1,3,6,2,5,4] => [3,7,4,6,5,2,1] => ? = 3
[1,2,4,7,5,3,6] => [2,6,3,5,7,4,1] => [7,1,3,6,4,2,5] => [3,7,5,4,6,2,1] => ? = 3
[1,2,4,7,5,6,3] => [2,7,3,5,6,4,1] => [7,1,3,6,4,5,2] => [3,6,5,4,7,2,1] => ? = 4
[1,2,4,7,6,3,5] => [2,6,3,7,5,4,1] => [7,1,3,6,5,2,4] => [3,5,7,4,6,2,1] => ? = 2
[1,2,4,7,6,5,3] => [2,7,3,6,5,4,1] => [7,1,3,6,5,4,2] => [3,5,6,4,7,2,1] => ? = 2
[1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => [7,1,4,2,3,5,6] => [7,6,5,3,4,2,1] => ? = 4
[1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => [7,1,4,2,3,6,5] => [6,7,5,3,4,2,1] => ? = 2
[1,2,5,3,6,4,7] => [2,4,6,3,5,7,1] => [7,1,4,2,5,3,6] => [5,7,6,3,4,2,1] => ? = 3
Description
The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form (e+1,e) in the word corresponding to the permutation in one-line notation. This statistic, adj0, counts adjacencies in the word with a zero appended.
(adj0,des) and (fix,exc) are equidistributed, see [1].
Matching statistic: St000894
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 54%
St000894: Alternating sign matrices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 54%
Values
[1] => [[1]]
=> ? = 1
[1,2] => [[1,0],[0,1]]
=> 2
[2,1] => [[0,1],[1,0]]
=> 0
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[1,2,3,4,5] => [[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]]
=> 5
[1,2,3,5,4] => [[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]]
=> 3
[1,2,4,3,5] => [[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]]
=> 3
[1,2,4,5,3] => [[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]]
=> 2
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 2
[1,2,5,4,3] => [[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]]
=> 3
[1,3,2,4,5] => [[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]]
=> 3
[1,3,2,5,4] => [[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
[1,3,4,2,5] => [[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]]
=> 2
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 1
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => [[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]]
=> 3
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 2
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3
[1,2,3,6,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3
[1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 4
[1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2
[1,2,4,6,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 2
[1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3
[1,2,5,3,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 2
[1,2,5,4,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 3
[1,2,5,6,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 2
[1,2,5,6,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 2
[1,2,6,3,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 2
[1,2,6,3,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 3
[1,2,6,4,3,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 3
[1,2,6,4,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
[1,2,6,5,3,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 2
[1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,3,4,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,4,6,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,3,4,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,5,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 2
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,5,6,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,3,6,2,5,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,3,6,4,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 2
[1,3,6,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 3
[1,3,6,5,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,4,2,5,6,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,4,2,6,5,3] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,4,3,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2
[1,4,3,6,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 2
[1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,5,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,4,5,6,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,4,6,2,5,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,6,3,2,5] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 1
[1,4,6,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,4,6,5,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,4,6,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,2,3,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
Description
The trace of an alternating sign matrix.
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