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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000022
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 2
[1,2,3] => [2,3,1] => [3,1,2] => 0
[1,3,2] => [2,1,3] => [2,1,3] => 1
[2,1,3] => [3,2,1] => [3,2,1] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [3,1,2] => [2,3,1] => 0
[3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 1
[1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 1
[1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2
[1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 0
[1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 1
[2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
[3,1,2,4] => [3,4,2,1] => [4,3,1,2] => 0
[3,1,4,2] => [3,1,2,4] => [2,3,1,4] => 1
[3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2
[3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 0
[3,4,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [3,1,4,2] => [2,4,1,3] => 0
[4,2,1,3] => [4,3,1,2] => [3,4,2,1] => 0
[4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,1,2] => [4,1,3,2] => [2,4,3,1] => 1
[4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[1,2,3,4,5] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [2,3,4,1,5] => [4,1,2,3,5] => 1
[1,2,4,3,5] => [2,3,5,4,1] => [5,1,2,4,3] => 1
[1,2,4,5,3] => [2,3,1,4,5] => [3,1,2,4,5] => 2
[1,2,5,3,4] => [2,3,5,1,4] => [4,1,2,5,3] => 0
[1,2,5,4,3] => [2,3,1,5,4] => [3,1,2,5,4] => 0
[1,3,2,4,5] => [2,4,3,5,1] => [5,1,3,2,4] => 1
[1,3,2,5,4] => [2,4,3,1,5] => [4,1,3,2,5] => 2
[1,3,4,2,5] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,3,4,5] => 3
[1,3,5,2,4] => [2,5,3,1,4] => [4,1,3,5,2] => 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,3,5,4] => 1
[1,4,2,3,5] => [2,4,5,3,1] => [5,1,4,2,3] => 0
[1,4,2,5,3] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[1,4,3,2,5] => [2,5,4,3,1] => [5,1,4,3,2] => 0
[1,4,3,5,2] => [2,1,4,3,5] => [2,1,4,3,5] => 1
[1,4,5,2,3] => [2,5,1,3,4] => [3,1,4,5,2] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000475
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,2] => [2,1] => [2]
=> 0
[2,1] => [1,2] => [1,1]
=> 2
[1,2,3] => [2,3,1] => [3]
=> 0
[1,3,2] => [2,1,3] => [2,1]
=> 1
[2,1,3] => [3,2,1] => [2,1]
=> 1
[2,3,1] => [1,2,3] => [1,1,1]
=> 3
[3,1,2] => [3,1,2] => [3]
=> 0
[3,2,1] => [1,3,2] => [2,1]
=> 1
[1,2,3,4] => [2,3,4,1] => [4]
=> 0
[1,2,4,3] => [2,3,1,4] => [3,1]
=> 1
[1,3,2,4] => [2,4,3,1] => [3,1]
=> 1
[1,3,4,2] => [2,1,3,4] => [2,1,1]
=> 2
[1,4,2,3] => [2,4,1,3] => [4]
=> 0
[1,4,3,2] => [2,1,4,3] => [2,2]
=> 0
[2,1,3,4] => [3,2,4,1] => [3,1]
=> 1
[2,1,4,3] => [3,2,1,4] => [2,1,1]
=> 2
[2,3,1,4] => [4,2,3,1] => [2,1,1]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 4
[2,4,1,3] => [4,2,1,3] => [3,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 2
[3,1,2,4] => [3,4,2,1] => [4]
=> 0
[3,1,4,2] => [3,1,2,4] => [3,1]
=> 1
[3,2,1,4] => [4,3,2,1] => [2,2]
=> 0
[3,2,4,1] => [1,3,2,4] => [2,1,1]
=> 2
[3,4,1,2] => [4,1,2,3] => [4]
=> 0
[3,4,2,1] => [1,4,2,3] => [3,1]
=> 1
[4,1,2,3] => [3,4,1,2] => [2,2]
=> 0
[4,1,3,2] => [3,1,4,2] => [4]
=> 0
[4,2,1,3] => [4,3,1,2] => [4]
=> 0
[4,2,3,1] => [1,3,4,2] => [3,1]
=> 1
[4,3,1,2] => [4,1,3,2] => [3,1]
=> 1
[4,3,2,1] => [1,4,3,2] => [2,1,1]
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => [5]
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [4,1]
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => [4,1]
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => [3,1,1]
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => [5]
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => [3,2]
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => [4,1]
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => [3,1,1]
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => [3,1,1]
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,1,1]
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => [4,1]
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,2,1]
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => [5]
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => [4,1]
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => [3,2]
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => [2,2,1]
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => [5]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000895
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1
[1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [1,2] => [[1,0],[0,1]]
=> 2
[1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 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
[1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[2,1,3,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[2,1,4,3] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[2,3,1,4] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[2,4,3,1] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[3,1,2,4] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[3,1,4,2] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[3,2,1,4] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[3,2,4,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[3,4,2,1] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[4,1,2,3] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[4,1,3,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[4,2,1,3] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[4,2,3,1] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[4,3,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,3,2,1] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 0
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)
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 2
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [2,1,3] => [2,1,3] => 1
[2,1,3] => [3,1,2] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,3,2] => [3,1,2] => [3,2,1] => 3
[3,1,2] => [2,1,3] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [2,3,1] => 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 1
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => [3,2,1,4] => 2
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [3,1,2,4] => 0
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 2
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => [4,3,2,1] => 4
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => [2,4,3,1] => 2
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => [1,4,2,3] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 0
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => [3,4,2,1] => 2
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [3,1,4,2] => 0
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => [4,2,3,1] => 1
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => [2,4,1,3] => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => [3,4,1,2] => 0
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [2,3,4,1] => 1
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => [4,1,3,2] => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [3,2,4,1] => 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => [3,2,1,4,5] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => [4,3,2,1,5] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => [2,4,3,1,5] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => [4,2,1,3,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => [3,4,2,1,5] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => [3,1,4,2,5] => 0
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 1
[4,3,2,1,7,6,5] => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => [2,3,1,6,5,7,4] => ? = 1
[5,4,3,2,1,7,6] => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => ? = 1
[7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1
[7,8,5,4,1,2,3,6] => [6,3,2,1,4,5,8,7] => [3,6,7,8,5,4,1,2] => [5,7,1,3,8,2,6,4] => ? = 0
[8,6,5,4,3,1,2,7] => [7,2,1,3,4,5,6,8] => [2,7,8,6,5,4,3,1] => [5,6,4,8,1,2,7,3] => ? = 0
[8,5,6,3,4,1,2,7] => [7,2,1,4,3,6,5,8] => [2,7,8,5,6,3,4,1] => [8,1,2,7,4,5,6,3] => ? = 0
[7,6,5,4,3,2,1,8] => [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,5,6,4,7,3,8,2] => ? = 0
[7,5,6,3,4,2,1,8] => [8,1,2,4,3,6,5,7] => [1,8,7,5,6,3,4,2] => [1,7,4,5,6,3,8,2] => ? = 0
[6,7,5,4,2,3,1,8] => [8,1,3,2,4,5,7,6] => [1,8,6,7,5,4,2,3] => [1,5,8,3,6,4,7,2] => ? = 0
[6,5,7,3,2,4,1,8] => [8,1,4,2,3,7,5,6] => [1,8,5,7,6,2,4,3] => [1,7,4,8,3,5,6,2] => ? = 0
[5,6,7,2,3,4,1,8] => [8,1,4,3,2,7,6,5] => [1,8,5,6,7,2,3,4] => [1,7,3,5,8,4,6,2] => ? = 0
[1,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => ? = 0
[1,4,3,8,7,6,5,2] => [2,5,6,7,8,3,4,1] => [7,4,3,2,1,6,5,8] => [3,4,2,6,7,5,1,8] => ? = 0
[1,6,5,4,3,8,7,2] => [2,7,8,3,4,5,6,1] => [7,2,1,6,5,4,3,8] => [2,5,6,4,7,3,1,8] => ? = 0
[1,4,3,6,5,8,7,2] => [2,7,8,5,6,3,4,1] => [7,2,1,4,3,6,5,8] => [2,4,6,7,5,3,1,8] => ? = 0
[3,2,1,8,7,6,5,4] => [4,5,6,7,8,1,2,3] => [5,4,3,2,1,8,7,6] => [3,4,2,5,1,7,8,6] => ? = 0
[3,2,1,6,5,8,7,4] => [4,7,8,5,6,1,2,3] => [5,2,1,4,3,8,7,6] => [2,4,5,3,1,7,8,6] => ? = 0
[5,4,3,2,1,8,7,6] => [6,7,8,1,2,3,4,5] => [3,2,1,8,7,6,5,4] => [2,3,1,6,7,5,8,4] => ? = 0
[3,2,5,4,1,8,7,6] => [6,7,8,1,4,5,2,3] => [3,2,1,8,5,4,7,6] => [2,3,1,5,7,8,6,4] => ? = 0
[3,2,7,6,5,4,1,8] => [8,1,4,5,6,7,2,3] => [1,8,5,4,3,2,7,6] => [1,4,5,3,7,8,6,2] => ? = 0
[5,4,3,2,7,6,1,8] => [8,1,6,7,2,3,4,5] => [1,8,3,2,7,6,5,4] => [1,3,6,7,5,8,4,2] => ? = 0
[1,8,5,4,7,6,3,2] => [2,3,6,7,4,5,8,1] => [7,6,3,2,5,4,1,8] => [3,5,6,4,2,7,1,8] => ? = 0
[7,4,3,6,5,2,1,8] => [8,1,2,5,6,3,4,7] => [1,8,7,4,3,6,5,2] => [1,4,6,7,5,3,8,2] => ? = 0
[8,7,6,5,4,3,2,1,10,9] => [9,10,1,2,3,4,5,6,7,8] => [2,1,10,9,8,7,6,5,4,3] => [2,1,7,6,8,5,9,4,10,3] => ? = 2
[10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => [6,5,7,4,8,3,9,2,10,1] => ? = 2
[3,2,7,8,1,4,5,6] => [6,5,4,1,8,7,2,3] => [3,4,5,8,1,2,7,6] => [5,1,3,7,8,6,2,4] => ? = 0
[7,4,3,8,1,2,5,6] => [6,5,2,1,8,3,4,7] => [3,4,7,8,1,6,5,2] => [6,7,5,1,3,8,2,4] => ? = 0
[5,4,3,2,8,1,6,7] => [7,6,1,8,2,3,4,5] => [2,3,8,1,7,6,5,4] => [6,7,5,8,4,1,2,3] => ? = 0
[4,5,2,3,7,6,1,8] => [8,1,6,7,3,2,5,4] => [1,8,3,2,6,7,4,5] => [1,3,8,5,6,7,4,2] => ? = 0
[6,4,3,7,2,5,1,8] => [8,1,5,2,7,3,4,6] => [1,8,4,7,2,6,5,3] => [1,6,8,3,4,7,5,2] => ? = 0
[8,4,3,6,5,1,2,7] => [7,2,1,5,6,3,4,8] => [2,7,8,4,3,6,5,1] => [4,6,8,1,2,7,5,3] => ? = 0
[1,5,6,3,4,8,7,2] => [2,7,8,4,3,6,5,1] => [7,2,1,5,6,3,4,8] => [2,7,4,5,6,3,1,8] => ? = 0
[4,6,2,7,3,5,1,8] => [8,1,5,3,7,2,6,4] => [1,8,4,6,2,7,3,5] => [1,7,3,4,6,8,5,2] => ? = 0
[3,2,6,7,4,5,1,8] => [8,1,5,4,7,6,2,3] => [1,8,4,5,2,3,7,6] => [1,7,8,6,3,4,5,2] => ? = 0
[4,7,2,6,5,3,1,8] => [8,1,3,5,6,2,7,4] => [1,8,6,4,3,7,2,5] => [1,4,8,5,3,6,7,2] => ? = 0
[5,7,6,2,4,3,1,8] => [8,1,3,4,2,6,7,5] => [1,8,6,5,7,3,2,4] => [1,6,3,8,4,5,7,2] => ? = 0
[1,7,5,4,8,3,6,2] => [2,6,3,8,4,5,7,1] => [7,3,6,1,5,4,2,8] => [5,7,2,3,6,4,1,8] => ? = 0
[8,7,5,4,1,3,2,6] => [6,2,3,1,4,5,7,8] => [3,7,6,8,5,4,2,1] => [5,7,2,8,1,3,6,4] => ? = 0
[6,8,5,4,2,1,3,7] => [7,3,1,2,4,5,8,6] => [2,6,8,7,5,4,1,3] => [5,7,1,2,6,4,8,3] => ? = 0
[5,8,6,2,4,1,3,7] => [7,3,1,4,2,6,8,5] => [2,6,8,5,7,3,1,4] => [8,4,5,7,1,2,6,3] => ? = 0
[4,8,2,6,5,1,3,7] => [7,3,1,5,6,2,8,4] => [2,6,8,4,3,7,1,5] => [4,7,1,2,6,8,5,3] => ? = 0
[3,2,8,6,5,1,4,7] => [7,4,1,5,6,8,2,3] => [2,5,8,4,3,1,7,6] => [4,7,8,6,1,2,5,3] => ? = 0
[4,1,2,6,5,8,7,3] => [3,7,8,5,6,2,1,4] => [6,2,1,4,3,7,8,5] => [2,4,8,5,3,1,6,7] => ? = 0
[5,1,6,2,4,8,7,3] => [3,7,8,4,2,6,1,5] => [6,2,1,5,7,3,8,4] => [2,6,3,1,8,4,5,7] => ? = 0
[6,1,5,4,2,8,7,3] => [3,7,8,2,4,5,1,6] => [6,2,1,7,5,4,8,3] => [2,5,8,3,1,6,4,7] => ? = 0
[7,1,5,4,8,2,6,3] => [3,6,2,8,4,5,1,7] => [6,3,7,1,5,4,8,2] => [5,6,4,1,8,2,3,7] => ? = 0
[8,1,5,4,7,6,2,3] => [3,2,6,7,4,5,1,8] => [6,7,3,2,5,4,8,1] => [3,5,8,1,6,4,2,7] => ? = 0
[1,8,6,7,4,5,3,2] => [2,3,5,4,7,6,8,1] => [7,6,4,5,2,3,1,8] => [6,3,4,5,2,7,1,8] => ? = 0
[8,1,6,7,4,5,2,3] => [3,2,5,4,7,6,1,8] => [6,7,4,5,2,3,8,1] => [8,1,6,3,4,5,2,7] => ? = 0
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, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended.
$(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000247
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> ? = 1
[1,2] => [2,1] => {{1,2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 2
[1,2,3] => [2,3,1] => {{1,2,3}}
=> 0
[1,3,2] => [2,1,3] => {{1,2},{3}}
=> 1
[2,1,3] => [3,2,1] => {{1,3},{2}}
=> 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> 3
[3,1,2] => [3,1,2] => {{1,2,3}}
=> 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,4,2,3] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> 1
[2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[3,1,2,4] => [3,4,2,1] => {{1,2,3,4}}
=> 0
[3,1,4,2] => [3,1,2,4] => {{1,2,3},{4}}
=> 1
[3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
[3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,4,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
[4,1,3,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,1,3] => [4,3,1,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,3,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => {{1,2,4,5},{3}}
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => {{1,2,3,4,5}}
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[1,4,5,3,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 2
[6,7,8,1,2,3,4,5] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> ? = 0
[7,8,5,4,1,2,3,6] => [6,7,8,5,4,1,2,3] => {{1,6},{2,7},{3,8},{4,5}}
=> ? = 0
[8,6,5,4,3,1,2,7] => [7,8,6,5,4,3,1,2] => {{1,7},{2,8},{3,6},{4,5}}
=> ? = 0
[8,5,6,3,4,1,2,7] => [7,8,5,6,3,4,1,2] => {{1,7},{2,8},{3,5},{4,6}}
=> ? = 0
[7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 0
[7,5,6,3,4,2,1,8] => [8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> ? = 0
[6,7,5,4,2,3,1,8] => [8,6,7,5,4,2,3,1] => {{1,8},{2,6},{3,7},{4,5}}
=> ? = 0
[6,5,7,3,2,4,1,8] => [8,6,5,7,3,2,4,1] => {{1,8},{2,6},{3,5},{4,7}}
=> ? = 0
[5,6,7,2,3,4,1,8] => [8,5,6,7,2,3,4,1] => {{1,8},{2,5},{3,6},{4,7}}
=> ? = 0
[1,8,7,6,5,4,3,2] => [2,1,8,7,6,5,4,3] => {{1,2},{3,8},{4,7},{5,6}}
=> ? = 0
[1,4,3,8,7,6,5,2] => [2,1,4,3,8,7,6,5] => {{1,2},{3,4},{5,8},{6,7}}
=> ? = 0
[1,6,5,4,3,8,7,2] => [2,1,6,5,4,3,8,7] => {{1,2},{3,6},{4,5},{7,8}}
=> ? = 0
[1,4,3,6,5,8,7,2] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 0
[3,2,1,8,7,6,5,4] => [4,3,2,1,8,7,6,5] => {{1,4},{2,3},{5,8},{6,7}}
=> ? = 0
[3,2,1,6,5,8,7,4] => [4,3,2,1,6,5,8,7] => {{1,4},{2,3},{5,6},{7,8}}
=> ? = 0
[5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,8,7] => {{1,6},{2,5},{3,4},{7,8}}
=> ? = 0
[3,2,5,4,1,8,7,6] => [6,3,2,5,4,1,8,7] => {{1,6},{2,3},{4,5},{7,8}}
=> ? = 0
[6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 2
[3,2,7,6,5,4,1,8] => [8,3,2,7,6,5,4,1] => {{1,8},{2,3},{4,7},{5,6}}
=> ? = 0
[5,4,3,2,7,6,1,8] => [8,5,4,3,2,7,6,1] => {{1,8},{2,5},{3,4},{6,7}}
=> ? = 0
[3,2,5,4,7,6,1,8] => [8,3,2,5,4,7,6,1] => {{1,8},{2,3},{4,5},{6,7}}
=> ? = 0
[1,8,5,4,7,6,3,2] => [2,1,8,5,4,7,6,3] => {{1,2},{3,8},{4,5},{6,7}}
=> ? = 0
[7,4,3,6,5,2,1,8] => [8,7,4,3,6,5,2,1] => {{1,8},{2,7},{3,4},{5,6}}
=> ? = 0
[8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => {{1,9},{2,8},{3,7},{4,6},{5},{10}}
=> ? = 2
[10,9,8,7,6,5,4,3,2,1] => [1,10,9,8,7,6,5,4,3,2] => {{1},{2,10},{3,9},{4,8},{5,7},{6}}
=> ? = 2
[3,2,7,8,1,4,5,6] => [6,3,2,7,8,1,4,5] => {{1,6},{2,3},{4,7},{5,8}}
=> ? = 0
[7,4,3,8,1,2,5,6] => [6,7,4,3,8,1,2,5] => {{1,6},{2,7},{3,4},{5,8}}
=> ? = 0
[5,4,3,2,8,1,6,7] => [7,5,4,3,2,8,1,6] => {{1,7},{2,5},{3,4},{6,8}}
=> ? = 0
[4,5,2,3,7,6,1,8] => [8,4,5,2,3,7,6,1] => {{1,8},{2,4},{3,5},{6,7}}
=> ? = 0
[6,4,3,7,2,5,1,8] => [8,6,4,3,7,2,5,1] => {{1,8},{2,6},{3,4},{5,7}}
=> ? = 0
[8,4,3,6,5,1,2,7] => [7,8,4,3,6,5,1,2] => {{1,7},{2,8},{3,4},{5,6}}
=> ? = 0
[1,5,6,3,4,8,7,2] => [2,1,5,6,3,4,8,7] => {{1,2},{3,5},{4,6},{7,8}}
=> ? = 0
[4,6,2,7,3,5,1,8] => [8,4,6,2,7,3,5,1] => {{1,8},{2,4},{3,6},{5,7}}
=> ? = 0
[3,2,6,7,4,5,1,8] => [8,3,2,6,7,4,5,1] => {{1,8},{2,3},{4,6},{5,7}}
=> ? = 0
[4,7,2,6,5,3,1,8] => [8,4,7,2,6,5,3,1] => {{1,8},{2,4},{3,7},{5,6}}
=> ? = 0
[5,7,6,2,4,3,1,8] => [8,5,7,6,2,4,3,1] => {{1,8},{2,5},{3,7},{4,6}}
=> ? = 0
[1,7,5,4,8,3,6,2] => [2,1,7,5,4,8,3,6] => {{1,2},{3,7},{4,5},{6,8}}
=> ? = 0
[8,7,5,4,1,3,2,6] => [6,8,7,5,4,1,3,2] => {{1,6},{2,8},{3,7},{4,5}}
=> ? = 0
[6,8,5,4,2,1,3,7] => [7,6,8,5,4,2,1,3] => {{1,7},{2,6},{3,8},{4,5}}
=> ? = 0
[5,8,6,2,4,1,3,7] => [7,5,8,6,2,4,1,3] => {{1,7},{2,5},{3,8},{4,6}}
=> ? = 0
[4,8,2,6,5,1,3,7] => [7,4,8,2,6,5,1,3] => {{1,7},{2,4},{3,8},{5,6}}
=> ? = 0
[3,2,8,6,5,1,4,7] => [7,3,2,8,6,5,1,4] => {{1,7},{2,3},{4,8},{5,6}}
=> ? = 0
[4,1,2,6,5,8,7,3] => [3,4,1,2,6,5,8,7] => {{1,3},{2,4},{5,6},{7,8}}
=> ? = 0
[5,1,6,2,4,8,7,3] => [3,5,1,6,2,4,8,7] => {{1,3},{2,5},{4,6},{7,8}}
=> ? = 0
[6,1,5,4,2,8,7,3] => [3,6,1,5,4,2,8,7] => {{1,3},{2,6},{4,5},{7,8}}
=> ? = 0
[7,1,5,4,8,2,6,3] => [3,7,1,5,4,8,2,6] => {{1,3},{2,7},{4,5},{6,8}}
=> ? = 0
[8,1,5,4,7,6,2,3] => [3,8,1,5,4,7,6,2] => {{1,3},{2,8},{4,5},{6,7}}
=> ? = 0
[1,8,6,7,4,5,3,2] => [2,1,8,6,7,4,5,3] => {{1,2},{3,8},{4,6},{5,7}}
=> ? = 0
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> {{1}}
=> ? = 1
[1,2] => [2,1] => {{1,2}}
=> {{1},{2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> {{1,2}}
=> 2
[1,2,3] => [2,3,1] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,3,2] => [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[2,1,3] => [3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> {{1,2,3}}
=> 3
[3,1,2] => [3,1,2] => {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,4,2,3] => [2,4,1,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
[2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 2
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
[2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[3,1,2,4] => [3,4,2,1] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,1,4,2] => [3,1,2,4] => {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 0
[3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[3,4,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[4,1,2,3] => [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
[4,1,3,2] => [3,1,4,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[4,2,1,3] => [4,3,1,2] => {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[4,3,1,2] => [4,1,3,2] => {{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
[4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => {{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[1,4,5,3,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
[8,7,6,5,4,3,2,1] => [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 2
[8,2,3,4,5,6,7,1] => [1,3,4,5,6,7,8,2] => {{1},{2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[6,7,8,1,2,3,4,5] => [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> {{1,5},{2,6},{3,7},{4,8}}
=> ? = 0
[7,8,5,4,1,2,3,6] => [6,7,8,5,4,1,2,3] => {{1,6},{2,7},{3,8},{4,5}}
=> {{1,4,6},{2,7},{3,8},{5}}
=> ? = 0
[8,6,5,4,3,1,2,7] => [7,8,6,5,4,3,1,2] => {{1,7},{2,8},{3,6},{4,5}}
=> {{1,3,7},{2,8},{4,6},{5}}
=> ? = 0
[8,5,6,3,4,1,2,7] => [7,8,5,6,3,4,1,2] => {{1,7},{2,8},{3,5},{4,6}}
=> {{1,7},{2,8},{3,5},{4,6}}
=> ? = 0
[7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 0
[7,5,6,3,4,2,1,8] => [8,7,5,6,3,4,2,1] => {{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,8},{3,5,7},{4,6}}
=> ? = 0
[6,7,5,4,2,3,1,8] => [8,6,7,5,4,2,3,1] => {{1,8},{2,6},{3,7},{4,5}}
=> {{1},{2,4,6,8},{3,7},{5}}
=> ? = 0
[6,5,7,3,2,4,1,8] => [8,6,5,7,3,2,4,1] => {{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5,8},{3,7},{4,6}}
=> ? = 0
[5,6,7,2,3,4,1,8] => [8,5,6,7,2,3,4,1] => {{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2,5,8},{3,6},{4,7}}
=> ? = 0
[1,8,7,6,5,4,3,2] => [2,1,8,7,6,5,4,3] => {{1,2},{3,8},{4,7},{5,6}}
=> {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 0
[1,4,3,8,7,6,5,2] => [2,1,4,3,8,7,6,5] => {{1,2},{3,4},{5,8},{6,7}}
=> {{1,5,7},{2,4},{3},{6},{8}}
=> ? = 0
[1,6,5,4,3,8,7,2] => [2,1,6,5,4,3,8,7] => {{1,2},{3,6},{4,5},{7,8}}
=> {{1,3,7},{2},{4,6},{5},{8}}
=> ? = 0
[1,4,3,6,5,8,7,2] => [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 0
[3,2,1,8,7,6,5,4] => [4,3,2,1,8,7,6,5] => {{1,4},{2,3},{5,8},{6,7}}
=> {{1,5},{2,4},{3},{6,8},{7}}
=> ? = 0
[3,2,1,6,5,8,7,4] => [4,3,2,1,6,5,8,7] => {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,5},{2},{4},{6,8},{7}}
=> ? = 0
[5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,8,7] => {{1,6},{2,5},{3,4},{7,8}}
=> {{1,3},{2},{4,8},{5,7},{6}}
=> ? = 0
[3,2,5,4,1,8,7,6] => [6,3,2,5,4,1,8,7] => {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3},{2},{4,6,8},{5},{7}}
=> ? = 0
[6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => {{1,7},{2,6},{3,5},{4},{8}}
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? = 2
[1,2,3,4,5,6,8,7] => [2,3,4,5,6,7,1,8] => {{1,2,3,4,5,6,7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[3,2,7,6,5,4,1,8] => [8,3,2,7,6,5,4,1] => {{1,8},{2,3},{4,7},{5,6}}
=> {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 0
[5,4,3,2,7,6,1,8] => [8,5,4,3,2,7,6,1] => {{1,8},{2,5},{3,4},{6,7}}
=> {{1},{2,4,8},{3},{5,7},{6}}
=> ? = 0
[3,2,5,4,7,6,1,8] => [8,3,2,5,4,7,6,1] => {{1,8},{2,3},{4,5},{6,7}}
=> {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 0
[1,8,5,4,7,6,3,2] => [2,1,8,5,4,7,6,3] => {{1,2},{3,8},{4,5},{6,7}}
=> {{1,7},{2,4,6},{3},{5},{8}}
=> ? = 0
[7,4,3,6,5,2,1,8] => [8,7,4,3,6,5,2,1] => {{1,8},{2,7},{3,4},{5,6}}
=> {{1},{2,8},{3,5,7},{4},{6}}
=> ? = 0
[8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => {{1,9},{2,8},{3,7},{4,6},{5},{10}}
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> ? = 2
[10,9,8,7,6,5,4,3,2,1] => [1,10,9,8,7,6,5,4,3,2] => {{1},{2,10},{3,9},{4,8},{5,7},{6}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ? = 2
[3,2,7,8,1,4,5,6] => [6,3,2,7,8,1,4,5] => {{1,6},{2,3},{4,7},{5,8}}
=> {{1,4},{2,5},{3,6,8},{7}}
=> ? = 0
[7,4,3,8,1,2,5,6] => [6,7,4,3,8,1,2,5] => {{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2,5,7},{3,8},{6}}
=> ? = 0
[5,4,3,2,8,1,6,7] => [7,5,4,3,2,8,1,6] => {{1,7},{2,5},{3,4},{6,8}}
=> {{1,3},{2,4,8},{5,7},{6}}
=> ? = 0
[4,5,2,3,7,6,1,8] => [8,4,5,2,3,7,6,1] => {{1,8},{2,4},{3,5},{6,7}}
=> {{1},{2,4,6,8},{3},{5,7}}
=> ? = 0
[6,4,3,7,2,5,1,8] => [8,6,4,3,7,2,5,1] => {{1,8},{2,6},{3,4},{5,7}}
=> {{1},{2,4,8},{3,5,7},{6}}
=> ? = 0
[8,4,3,6,5,1,2,7] => [7,8,4,3,6,5,1,2] => {{1,7},{2,8},{3,4},{5,6}}
=> {{1,3,5,7},{2,8},{4},{6}}
=> ? = 0
[1,5,6,3,4,8,7,2] => [2,1,5,6,3,4,8,7] => {{1,2},{3,5},{4,6},{7,8}}
=> {{1,3,5,7},{2},{4,6},{8}}
=> ? = 0
[4,6,2,7,3,5,1,8] => [8,4,6,2,7,3,5,1] => {{1,8},{2,4},{3,6},{5,7}}
=> {{1},{2,4,8},{3,6},{5,7}}
=> ? = 0
[3,2,6,7,4,5,1,8] => [8,3,2,6,7,4,5,1] => {{1,8},{2,3},{4,6},{5,7}}
=> {{1},{2,4,6,8},{3,5},{7}}
=> ? = 0
[4,7,2,6,5,3,1,8] => [8,4,7,2,6,5,3,1] => {{1,8},{2,4},{3,7},{5,6}}
=> {{1},{2,6,8},{3,5,7},{4}}
=> ? = 0
[5,7,6,2,4,3,1,8] => [8,5,7,6,2,4,3,1] => {{1,8},{2,5},{3,7},{4,6}}
=> {{1},{2,6,8},{3,5},{4,7}}
=> ? = 0
[1,7,5,4,8,3,6,2] => [2,1,7,5,4,8,3,6] => {{1,2},{3,7},{4,5},{6,8}}
=> {{1,3,7},{2,4,6},{5},{8}}
=> ? = 0
[8,7,5,4,1,3,2,6] => [6,8,7,5,4,1,3,2] => {{1,6},{2,8},{3,7},{4,5}}
=> {{1,7},{2,4,6},{3,8},{5}}
=> ? = 0
[6,8,5,4,2,1,3,7] => [7,6,8,5,4,2,1,3] => {{1,7},{2,6},{3,8},{4,5}}
=> {{1,4,6},{2,8},{3,7},{5}}
=> ? = 0
[5,8,6,2,4,1,3,7] => [7,5,8,6,2,4,1,3] => {{1,7},{2,5},{3,8},{4,6}}
=> {{1,6},{2,8},{3,5},{4,7}}
=> ? = 0
[4,8,2,6,5,1,3,7] => [7,4,8,2,6,5,1,3] => {{1,7},{2,4},{3,8},{5,6}}
=> {{1,6},{2,8},{3,5,7},{4}}
=> ? = 0
[3,2,8,6,5,1,4,7] => [7,3,2,8,6,5,1,4] => {{1,7},{2,3},{4,8},{5,6}}
=> {{1,3,5},{2,6,8},{4},{7}}
=> ? = 0
[4,1,2,6,5,8,7,3] => [3,4,1,2,6,5,8,7] => {{1,3},{2,4},{5,6},{7,8}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> ? = 0
[5,1,6,2,4,8,7,3] => [3,5,1,6,2,4,8,7] => {{1,3},{2,5},{4,6},{7,8}}
=> {{1,3,5},{2},{4,7},{6,8}}
=> ? = 0
[6,1,5,4,2,8,7,3] => [3,6,1,5,4,2,8,7] => {{1,3},{2,6},{4,5},{7,8}}
=> {{1,3,7},{2},{4,6,8},{5}}
=> ? = 0
[7,1,5,4,8,2,6,3] => [3,7,1,5,4,8,2,6] => {{1,3},{2,7},{4,5},{6,8}}
=> {{1,3},{2,7},{4,6,8},{5}}
=> ? = 0
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: St000241
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000241: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 0
[2,1] => 2
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 4
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 2
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
[1,2,3,4,5,7,6] => ? = 1
[1,2,3,4,7,6,5] => ? = 0
[1,7,3,4,5,6,2] => ? = 0
[1,7,6,5,4,3,2] => ? = 1
[4,3,2,1,7,6,5] => ? = 1
[5,4,3,2,1,7,6] => ? = 1
[7,2,3,4,5,6,1] => ? = 1
[7,6,5,4,3,2,1] => ? = 1
[8,7,6,5,4,3,2,1] => ? = 2
[8,2,3,4,5,6,7,1] => ? = 1
[6,7,8,1,2,3,4,5] => ? = 0
[7,8,5,4,1,2,3,6] => ? = 0
[8,6,5,4,3,1,2,7] => ? = 0
[8,5,6,3,4,1,2,7] => ? = 0
[7,6,5,4,3,2,1,8] => ? = 0
[7,5,6,3,4,2,1,8] => ? = 0
[6,7,5,4,2,3,1,8] => ? = 0
[6,5,7,3,2,4,1,8] => ? = 0
[5,6,7,2,3,4,1,8] => ? = 0
[1,8,7,6,5,4,3,2] => ? = 0
[1,4,3,8,7,6,5,2] => ? = 0
[1,6,5,4,3,8,7,2] => ? = 0
[1,4,3,6,5,8,7,2] => ? = 0
[3,2,1,8,7,6,5,4] => ? = 0
[3,2,1,6,5,8,7,4] => ? = 0
[5,4,3,2,1,8,7,6] => ? = 0
[3,2,5,4,1,8,7,6] => ? = 0
[6,5,4,3,2,1,8,7] => ? = 2
[1,2,3,4,5,6,8,7] => ? = 1
[3,2,7,6,5,4,1,8] => ? = 0
[5,4,3,2,7,6,1,8] => ? = 0
[3,2,5,4,7,6,1,8] => ? = 0
[1,8,5,4,7,6,3,2] => ? = 0
[7,4,3,6,5,2,1,8] => ? = 0
[8,7,6,5,4,3,2,1,10,9] => ? = 2
[10,9,8,7,6,5,4,3,2,1] => ? = 2
[3,2,7,8,1,4,5,6] => ? = 0
[7,4,3,8,1,2,5,6] => ? = 0
[5,4,3,2,8,1,6,7] => ? = 0
[4,5,2,3,7,6,1,8] => ? = 0
[6,4,3,7,2,5,1,8] => ? = 0
[8,4,3,6,5,1,2,7] => ? = 0
[1,5,6,3,4,8,7,2] => ? = 0
[4,6,2,7,3,5,1,8] => ? = 0
[3,2,6,7,4,5,1,8] => ? = 0
[4,7,2,6,5,3,1,8] => ? = 0
[5,7,6,2,4,3,1,8] => ? = 0
[1,7,5,4,8,3,6,2] => ? = 0
[8,7,5,4,1,3,2,6] => ? = 0
[6,8,5,4,2,1,3,7] => ? = 0
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000894
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> ? = 1
[1,2] => [2,1] => [[0,1],[1,0]]
=> 0
[2,1] => [1,2] => [[1,0],[0,1]]
=> 2
[1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[2,1,3] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[3,1,2] => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 0
[3,2,1] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 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
[1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 1
[1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 0
[1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[2,1,3,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 1
[2,1,4,3] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[2,3,1,4] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 2
[2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[2,4,1,3] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1
[2,4,3,1] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[3,1,2,4] => [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 0
[3,1,4,2] => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 1
[3,2,1,4] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[3,2,4,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 0
[3,4,2,1] => [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 1
[4,1,2,3] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[4,1,3,2] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 0
[4,2,1,3] => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 0
[4,2,3,1] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[4,3,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1
[4,3,2,1] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2
[1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[1,2,5,3,4] => [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 0
[1,2,5,4,3] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,3,2,4,5] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,3,2,5,4] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 2
[1,3,4,2,5] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 2
[1,3,4,5,2] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[1,3,5,2,4] => [2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 1
[1,3,5,4,2] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,4,2,3,5] => [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 0
[1,4,2,5,3] => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[1,4,3,2,5] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 0
[1,4,3,5,2] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,4,5,2,3] => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 0
[1,4,5,3,2] => [2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 0
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,2,5,3,4,6] => [2,3,5,6,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,2,5,4,3,6] => [2,3,6,5,4,1] => [[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],[0,0,1,0,0,0]]
=> ? = 0
[1,2,5,6,3,4] => [2,3,6,1,4,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,2,5,6,4,3] => [2,3,1,6,4,5] => [[0,0,1,0,0,0],[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
[1,2,6,3,4,5] => [2,3,5,6,1,4] => [[0,0,0,0,1,0],[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
[1,2,6,3,5,4] => [2,3,5,1,6,4] => [[0,0,0,1,0,0],[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
[1,2,6,4,5,3] => [2,3,1,5,6,4] => [[0,0,1,0,0,0],[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
[1,2,6,5,4,3] => [2,3,1,6,5,4] => [[0,0,1,0,0,0],[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]]
=> ? = 1
[1,3,2,4,5,6] => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,2,5,4,6] => [2,4,3,6,5,1] => [[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],[0,0,0,1,0,0]]
=> ? = 2
[1,3,4,2,5,6] => [2,5,3,4,6,1] => [[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],[0,0,0,0,1,0]]
=> ? = 2
[1,3,4,5,2,6] => [2,6,3,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 3
[1,3,5,4,2,6] => [2,6,3,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1
[1,3,5,6,4,2] => [2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 1
[1,3,6,4,5,2] => [2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,3,6,5,4,2] => [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2
[1,4,2,3,5,6] => [2,4,5,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,4,3,2,5,6] => [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,4,3,5,2,6] => [2,6,4,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 1
[1,4,5,6,2,3] => [2,6,1,3,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 0
[1,4,5,6,3,2] => [2,1,6,3,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,4,6,2,5,3] => [2,5,1,3,6,4] => [[0,0,1,0,0,0],[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
[1,4,6,3,5,2] => [2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,5,2,3,4,6] => [2,4,5,6,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,5,2,4,3,6] => [2,4,6,5,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
[1,5,2,6,3,4] => [2,4,6,1,3,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 0
[1,5,2,6,4,3] => [2,4,1,6,3,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,5,3,2,4,6] => [2,5,4,6,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,5,3,4,2,6] => [2,6,4,5,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 0
[1,5,3,6,4,2] => [2,1,4,6,3,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 0
[1,5,4,2,3,6] => [2,5,6,4,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1
[1,5,4,3,2,6] => [2,6,5,4,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1
[1,5,6,2,3,4] => [2,5,6,1,3,4] => [[0,0,0,1,0,0],[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
[1,5,6,2,4,3] => [2,5,1,6,3,4] => [[0,0,1,0,0,0],[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
[1,5,6,3,4,2] => [2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,6,2,3,4,5] => [2,4,5,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,6,2,3,5,4] => [2,4,5,1,6,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,6,2,4,5,3] => [2,4,1,5,6,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 0
[1,6,3,4,5,2] => [2,1,4,5,6,3] => [[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],[0,0,0,0,1,0]]
=> ? = 0
[1,6,3,5,4,2] => [2,1,4,6,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 1
[1,6,4,3,5,2] => [2,1,5,4,6,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1
[1,6,4,5,3,2] => [2,1,6,4,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 2
[1,6,5,3,4,2] => [2,1,5,6,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0
[1,6,5,4,3,2] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0
Description
The trace of an alternating sign matrix.
Matching statistic: St001903
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001903: Parking functions ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 71%
Mp00305: Permutations —parking function⟶ Parking functions
St001903: Parking functions ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => 1
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 2
[1,2,3] => [2,3,1] => [2,3,1] => 0
[1,3,2] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 3
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 0
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 1
[1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 1
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 2
[1,4,2,3] => [3,4,2,1] => [3,4,2,1] => 0
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => 0
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 2
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => 0
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 1
[4,3,1,2] => [4,2,1,3] => [4,2,1,3] => 1
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 2
[1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
[1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 1
[1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 1
[1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
[1,2,5,3,4] => [2,4,5,3,1] => [2,4,5,3,1] => ? = 0
[1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 0
[1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
[1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
[1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => ? = 1
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 1
[1,4,2,3,5] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1
[1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 0
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 0
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0
[1,5,2,3,4] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 0
[1,5,2,4,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 0
[1,5,3,2,4] => [4,3,5,2,1] => [4,3,5,2,1] => ? = 0
[1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 0
[1,5,4,2,3] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 1
[2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => ? = 2
[2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => ? = 2
[2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => ? = 3
[2,1,5,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => ? = 1
[2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => ? = 1
[2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 2
[2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => ? = 3
[2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 2
[2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 3
[2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => ? = 1
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 2
[2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => ? = 1
[2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 3
[2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 1
[2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 1
[2,5,1,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 1
[2,5,3,1,4] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 1
[2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 2
[2,5,4,1,3] => [1,5,3,2,4] => [1,5,3,2,4] => ? = 2
[2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => ? = 3
[3,1,2,4,5] => [3,1,4,5,2] => [3,1,4,5,2] => ? = 0
[3,1,2,5,4] => [3,1,5,4,2] => [3,1,5,4,2] => ? = 1
Description
The number of fixed points of a parking function.
If $(a_1,\dots,a_n)$ is a parking function, a fixed point is an index $i$ such that $a_i = i$.
It can be shown [1] that the generating function for parking functions with respect to this statistic is
$$
\frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right).
$$
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