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Matching statistic: St000804
St000804: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 6
[1,2,3,5,4] => 5
[1,2,4,3,5] => 5
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 5
[1,3,2,5,4] => 4
[1,3,4,2,5] => 4
[1,3,4,5,2] => 3
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 1
Description
The number of occurrences of the vincular pattern |123 in a permutation.
This is the number of occurrences of the pattern $(1,2,3)$, such that the letter matched by $1$ is the first entry of the permutation.
Matching statistic: St000803
Mp00064: Permutations —reverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000803: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [3,2,1] => [1,3,2] => [1,3,2] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[2,3,1] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[3,1,2] => [2,1,3] => [3,2,1] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [3,1,2] => 0
[1,2,3,4] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3
[1,2,4,3] => [3,4,2,1] => [1,4,2,3] => [1,3,4,2] => 2
[1,3,2,4] => [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 2
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,2,3] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [4,3,1,2] => [4,1,3,2] => [2,4,3,1] => 1
[2,1,4,3] => [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 0
[2,3,1,4] => [4,1,3,2] => [3,1,4,2] => [2,4,1,3] => 1
[2,3,4,1] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[2,4,1,3] => [3,1,4,2] => [3,1,2,4] => [2,3,1,4] => 0
[2,4,3,1] => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
[3,1,2,4] => [4,2,1,3] => [4,3,1,2] => [3,4,2,1] => 0
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 0
[3,2,1,4] => [4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 0
[3,2,4,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 0
[3,4,1,2] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 0
[3,4,2,1] => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 0
[4,1,2,3] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,3,2] => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 0
[4,2,1,3] => [3,1,2,4] => [3,4,2,1] => [4,3,1,2] => 0
[4,2,3,1] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 0
[4,3,1,2] => [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,2,3] => [1,4,5,3,2] => 5
[1,2,4,3,5] => [5,3,4,2,1] => [1,5,3,4,2] => [1,5,3,4,2] => 5
[1,2,4,5,3] => [3,5,4,2,1] => [1,5,2,4,3] => [1,3,5,4,2] => 4
[1,2,5,3,4] => [4,3,5,2,1] => [1,5,3,2,4] => [1,4,3,5,2] => 4
[1,2,5,4,3] => [3,4,5,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,4,5,3,2] => [1,5,4,2,3] => 5
[1,3,2,5,4] => [4,5,2,3,1] => [1,4,5,2,3] => [1,4,5,2,3] => 4
[1,3,4,2,5] => [5,2,4,3,1] => [1,3,5,4,2] => [1,5,2,4,3] => 4
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [1,3,5,2,4] => [1,4,2,5,3] => 3
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,4,3,5,2] => [1,5,3,2,4] => 4
[1,4,2,5,3] => [3,5,2,4,1] => [1,4,2,5,3] => [1,3,5,2,4] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,4,5,2] => [1,5,2,3,4] => 3
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 15
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,7,6,5,4,2,3] => [1,6,7,5,4,3,2] => ? = 14
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,7,6,5,3,4,2] => [1,7,5,6,4,3,2] => ? = 14
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,7,6,5,2,4,3] => [1,5,7,6,4,3,2] => ? = 13
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,7,6,5,3,2,4] => [1,6,5,7,4,3,2] => ? = 13
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,7,6,5,2,3,4] => [1,5,6,7,4,3,2] => ? = 12
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,7,6,4,5,3,2] => [1,7,6,4,5,3,2] => ? = 14
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,7,6,4,5,2,3] => [1,6,7,4,5,3,2] => ? = 13
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,7,6,3,5,4,2] => [1,7,4,6,5,3,2] => ? = 13
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,7,6,2,5,4,3] => [1,4,7,6,5,3,2] => ? = 12
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,7,6,3,5,2,4] => [1,6,4,7,5,3,2] => ? = 12
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,7,6,2,5,3,4] => [1,4,6,7,5,3,2] => ? = 11
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,7,6,4,3,5,2] => [1,7,5,4,6,3,2] => ? = 13
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,7,6,4,2,5,3] => [1,5,7,4,6,3,2] => ? = 12
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,7,6,3,4,5,2] => [1,7,4,5,6,3,2] => ? = 12
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,6,2,4,5,3] => [1,4,7,5,6,3,2] => ? = 11
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,7,6,3,2,5,4] => [1,5,4,7,6,3,2] => ? = 11
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,7,6,4,3,2,5] => [1,6,5,4,7,3,2] => ? = 12
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 11
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,7,6,3,4,2,5] => [1,6,4,5,7,3,2] => ? = 11
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,7,6,2,4,3,5] => [1,4,6,5,7,3,2] => ? = 10
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,7,6,3,2,4,5] => [1,5,4,6,7,3,2] => ? = 10
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,7,5,6,4,3,2] => [1,7,6,5,3,4,2] => ? = 14
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,7,5,6,4,2,3] => [1,6,7,5,3,4,2] => ? = 13
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,7,5,6,3,4,2] => [1,7,5,6,3,4,2] => ? = 13
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,7,5,6,2,4,3] => [1,5,7,6,3,4,2] => ? = 12
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,7,5,6,3,2,4] => [1,6,5,7,3,4,2] => ? = 12
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,7,5,6,2,3,4] => [1,5,6,7,3,4,2] => ? = 11
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,7,4,6,5,3,2] => [1,7,6,3,5,4,2] => ? = 13
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,7,4,6,5,2,3] => [1,6,7,3,5,4,2] => ? = 12
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,7,3,6,5,4,2] => [1,7,3,6,5,4,2] => ? = 12
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,7,3,6,5,2,4] => [1,6,3,7,5,4,2] => ? = 11
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,7,4,6,3,5,2] => [1,7,5,3,6,4,2] => ? = 12
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [1,7,4,6,2,5,3] => [1,5,7,3,6,4,2] => ? = 11
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,7,3,6,4,5,2] => [1,7,3,5,6,4,2] => ? = 11
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,7,3,6,2,5,4] => [1,5,3,7,6,4,2] => ? = 10
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,7,4,6,3,2,5] => [1,6,5,3,7,4,2] => ? = 11
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,7,4,6,2,3,5] => [1,5,6,3,7,4,2] => ? = 10
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,7,3,6,4,2,5] => [1,6,3,5,7,4,2] => ? = 10
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,7,3,6,2,4,5] => [1,5,3,6,7,4,2] => ? = 9
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,7,5,4,6,3,2] => [1,7,6,4,3,5,2] => ? = 13
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,7,5,4,6,2,3] => [1,6,7,4,3,5,2] => ? = 12
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [1,7,5,3,6,4,2] => [1,7,4,6,3,5,2] => ? = 12
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [1,7,5,2,6,4,3] => [1,4,7,6,3,5,2] => ? = 11
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [1,7,5,3,6,2,4] => [1,6,4,7,3,5,2] => ? = 11
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,7,5,2,6,3,4] => [1,4,6,7,3,5,2] => ? = 10
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [1,7,4,5,6,3,2] => [1,7,6,3,4,5,2] => ? = 12
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [1,7,4,5,6,2,3] => [1,6,7,3,4,5,2] => ? = 11
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [1,7,3,5,6,4,2] => [1,7,3,6,4,5,2] => ? = 11
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [1,7,3,5,6,2,4] => [1,6,3,7,4,5,2] => ? = 10
Description
The number of occurrences of the vincular pattern |132 in a permutation.
This is the number of occurrences of the pattern $(1,3,2)$, such that the letter matched by $1$ is the first entry of the permutation.
Matching statistic: St000802
Mp00069: Permutations —complement⟶ Permutations
St000802: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 73%●distinct values known / distinct values provided: 69%
St000802: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 73%●distinct values known / distinct values provided: 69%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 1
[1,3,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 3
[1,2,4,3] => [4,3,1,2] => 2
[1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => 1
[2,4,1,3] => [3,1,4,2] => 0
[2,4,3,1] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => 0
[3,1,4,2] => [2,4,1,3] => 0
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => 0
[4,3,1,2] => [1,2,4,3] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 6
[1,2,3,5,4] => [5,4,3,1,2] => 5
[1,2,4,3,5] => [5,4,2,3,1] => 5
[1,2,4,5,3] => [5,4,2,1,3] => 4
[1,2,5,3,4] => [5,4,1,3,2] => 4
[1,2,5,4,3] => [5,4,1,2,3] => 3
[1,3,2,4,5] => [5,3,4,2,1] => 5
[1,3,2,5,4] => [5,3,4,1,2] => 4
[1,3,4,2,5] => [5,3,2,4,1] => 4
[1,3,4,5,2] => [5,3,2,1,4] => 3
[1,3,5,2,4] => [5,3,1,4,2] => 3
[1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,3,5,2] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [5,2,1,4,3] => 2
[1,4,5,3,2] => [5,2,1,3,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 15
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 14
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 14
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 13
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 13
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 12
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 14
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 13
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 13
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 12
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 12
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 11
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 13
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 12
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 12
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 11
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 11
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 10
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 12
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 11
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 11
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 10
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 10
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 9
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 14
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 13
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 13
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 12
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 12
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 11
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 13
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 12
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 12
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 11
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 11
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 10
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 12
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 11
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 11
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 10
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 10
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 9
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 11
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 10
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 10
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 9
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 9
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 8
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 13
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 12
Description
The number of occurrences of the vincular pattern |321 in a permutation.
This is the number of occurrences of the pattern $(3,2,1)$, such that the letter matched by $3$ is the first entry of the permutation.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 6%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 6%
Values
[1,2] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[1,3,2] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,3] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,2,1] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 3 + 1
[1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,3,4,2] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[1,4,2,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([],1)
=> ? = 1 + 1
[1,4,3,2] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,3,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1,4,3] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,3,1,4] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,4,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([],1)
=> ? = 1 + 1
[2,4,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[2,4,3,1] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,1,2,4] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,4,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,4,1] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,4,1,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[3,4,2,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,1,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,3,1,2] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1)],2)
=> ? = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 6 + 1
[1,2,3,5,4] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 5 + 1
[1,2,4,3,5] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 5 + 1
[1,2,4,5,3] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 4 + 1
[1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 4 + 1
[1,2,5,4,3] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1)],2)
=> ? = 3 + 1
[1,3,2,4,5] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1)],2)
=> ? = 5 + 1
[1,3,2,5,4] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 4 + 1
[1,3,4,2,5] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 4 + 1
[1,3,4,5,2] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 3 + 1
[1,3,5,2,4] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[1,3,5,4,2] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,4,2,3,5] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 4 + 1
[1,4,2,5,3] => [2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 + 1
[1,4,3,2,5] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1)],2)
=> ? = 3 + 1
[1,4,3,5,2] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 2 + 1
[1,4,5,2,3] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 2 + 1
[1,4,5,3,2] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([],1)
=> ? = 1 + 1
[1,5,2,3,4] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 3 + 1
[1,5,2,4,3] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,1)],2)
=> ? = 2 + 1
[1,5,3,2,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([],1)
=> ? = 2 + 1
[1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ([],1)
=> ? = 1 + 1
[1,5,4,2,3] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ([],1)
=> ? = 1 + 1
[1,5,4,3,2] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1)],2)
=> ? = 0 + 1
[2,1,3,4,5] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1)],2)
=> ? = 3 + 1
[2,1,3,5,4] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1)],2)
=> ? = 2 + 1
[4,1,2,3,5] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2,5] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3,5] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,3,1,5] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2,5] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,2,1,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,2,4,3] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,4,3,2] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,4,3] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,4,3,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,1,3,2] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,2,3,4,6] => [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,2,4,3,6] => [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,2,4,6] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,3,4,2,6] => [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,4,2,3,6] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,1,4,3,2,6] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,3,4,6] => [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,1,4,3,6] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,3,1,4,6] => [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,4,1,3,6] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,1,2,4,6] => [6,4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,1,4,2,6] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,2,1,4,6] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,4,1,2,6] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,3,4,2,1,6] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,1,2,3,6] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,1,3,2,6] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,2,1,3,6] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,2,3,1,6] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[5,4,3,1,2,6] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,1,2,3,5,4] => [1,3,4,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,1,2,5,4,3] => [1,3,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,1,3,2,5,4] => [1,3,5,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,1,3,5,4,2] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,1,5,2,4,3] => [1,3,2,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,1,5,4,3,2] => [1,3,2,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[6,2,1,3,5,4] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001491
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[1,2] => [1,2] => [1,2] => 1 => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => 0 => ? = 0 + 1
[1,2,3] => [1,3,2] => [3,1,2] => 00 => ? = 1 + 1
[1,3,2] => [1,3,2] => [3,1,2] => 00 => ? = 0 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 01 => 1 = 0 + 1
[2,3,1] => [2,3,1] => [2,3,1] => 00 => ? = 0 + 1
[3,1,2] => [3,1,2] => [1,3,2] => 10 => 1 = 0 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 00 => ? = 0 + 1
[1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 000 => ? = 3 + 1
[1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 000 => ? = 2 + 1
[1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 000 => ? = 2 + 1
[1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 000 => ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => [4,3,1,2] => 000 => ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 000 => ? = 0 + 1
[2,1,3,4] => [2,1,4,3] => [2,4,1,3] => 000 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 000 => ? = 0 + 1
[2,3,1,4] => [2,4,1,3] => [4,2,1,3] => 000 => ? = 1 + 1
[2,3,4,1] => [2,4,3,1] => [4,2,3,1] => 000 => ? = 1 + 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 000 => ? = 0 + 1
[2,4,3,1] => [2,4,3,1] => [4,2,3,1] => 000 => ? = 0 + 1
[3,1,2,4] => [3,1,4,2] => [1,3,4,2] => 100 => 1 = 0 + 1
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => 100 => 1 = 0 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 001 => 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 000 => ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 000 => ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 000 => ? = 0 + 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 000 => ? = 0 + 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 000 => ? = 0 + 1
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => 010 => 1 = 0 + 1
[4,2,3,1] => [4,2,3,1] => [2,4,3,1] => 000 => ? = 0 + 1
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 100 => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 000 => ? = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 6 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 5 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 5 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 4 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 4 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 3 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 5 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 4 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 4 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 3 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 3 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 2 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 4 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 3 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 3 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 1 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 3 + 1
[1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 1 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 1 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0000 => ? = 0 + 1
[2,1,3,4,5] => [2,1,5,4,3] => [5,2,4,1,3] => 0000 => ? = 3 + 1
[2,1,3,5,4] => [2,1,5,4,3] => [5,2,4,1,3] => 0000 => ? = 2 + 1
[4,2,1,3,5] => [4,2,1,5,3] => [2,1,4,5,3] => 0100 => 1 = 0 + 1
[4,2,1,5,3] => [4,2,1,5,3] => [2,1,4,5,3] => 0100 => 1 = 0 + 1
[4,3,1,2,5] => [4,3,1,5,2] => [1,4,3,5,2] => 1000 => 1 = 0 + 1
[4,3,1,5,2] => [4,3,1,5,2] => [1,4,3,5,2] => 1000 => 1 = 0 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 0001 => 1 = 0 + 1
[4,5,1,2,3] => [4,5,1,3,2] => [1,4,5,3,2] => 1000 => 1 = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [1,4,5,3,2] => 1000 => 1 = 0 + 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,4,2] => 1000 => 1 = 0 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [1,3,5,4,2] => 1000 => 1 = 0 + 1
[5,3,2,1,4] => [5,3,2,1,4] => [3,2,1,5,4] => 0010 => 1 = 0 + 1
[5,4,2,1,3] => [5,4,2,1,3] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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