Processing math: 100%

Your data matches 26 different statistics following compositions of up to 3 maps.
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Matching statistic: St000802
Mapping
St000802: 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] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
[1,4,5,3,2] => 0
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: St000801
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000801: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
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] => 0
[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] => 1
[1,2,3,4] => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 0
[1,2,4,3] => [3,4,2,1] => [1,4,2,3] => [1,3,4,2] => 0
[1,3,2,4] => [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 0
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[1,4,2,3] => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
[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] => 0
[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] => 0
[2,3,4,1] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[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] => 1
[3,2,4,1] => [1,4,2,3] => [2,4,1,3] => [3,1,4,2] => 1
[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] => 1
[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] => 1
[4,2,1,3] => [3,1,2,4] => [3,4,2,1] => [4,3,1,2] => 1
[4,2,3,1] => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 2
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,2,3] => [1,4,5,3,2] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [1,5,2,4,3] => [1,3,5,4,2] => 0
[1,2,5,3,4] => [4,3,5,2,1] => [1,5,3,2,4] => [1,4,3,5,2] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,4,5,3,2] => [1,5,4,2,3] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,3,5,4,2] => [1,5,2,4,3] => 0
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,3,5,2,4] => [1,4,2,5,3] => 0
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,4,3,5,2] => [1,5,3,2,4] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [1,4,2,5,3] => [1,3,5,2,4] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
[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] => ? = 0
Description
The number of occurrences of the vincular pattern |312 in a permutation. This is the number of occurrences of the pattern (3,1,2), such that the letter matched by 3 is the first entry of the permutation.
Matching statistic: St000804
Mapping
Mp00069: Permutations —complement⟶ Permutations
St000804: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[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] => 1
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [4,2,1,3] => 0
[1,4,2,3] => [4,1,3,2] => 0
[1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => 0
[2,3,4,1] => [3,2,1,4] => 0
[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] => 1
[3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,3,4,2] => 1
[4,2,3,1] => [1,3,2,4] => 2
[4,3,1,2] => [1,2,4,3] => 2
[4,3,2,1] => [1,2,3,4] => 3
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [5,4,2,1,3] => 0
[1,2,5,3,4] => [5,4,1,3,2] => 0
[1,2,5,4,3] => [5,4,1,2,3] => 0
[1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [5,3,2,1,4] => 0
[1,3,5,2,4] => [5,3,1,4,2] => 0
[1,3,5,4,2] => [5,3,1,2,4] => 0
[1,4,2,3,5] => [5,2,4,3,1] => 0
[1,4,2,5,3] => [5,2,4,1,3] => 0
[1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,3,5,2] => [5,2,3,1,4] => 0
[1,4,5,2,3] => [5,2,1,4,3] => 0
[1,4,5,3,2] => [5,2,1,3,4] => 0
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 0
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 0
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 0
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => ? = 0
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 0
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => ? = 0
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 0
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 0
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 0
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 0
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 0
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 0
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 0
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 0
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 0
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 0
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 0
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 0
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 0
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 0
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 0
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 0
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 0
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 0
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 0
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 0
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 0
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => ? = 0
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 0
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 0
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 0
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 0
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 0
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 0
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 0
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 0
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 0
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 0
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 0
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 0
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 0
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 0
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 0
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 0
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 0
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 0
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 0
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 0
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: St000260
(load all 3 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 40%●distinct values known / distinct values provided: 9%
Values
[1,2] => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => [1,2] => ([],2)
 => ? = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 1
[2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 1
[3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => ? = 0 + 1
[3,2,1] => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => ? = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 0 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ? = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => ? = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => ? = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,5,3,1,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 0 + 1
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 1
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,5,6,3] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,4,6,3] => [.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000259
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 9%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(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] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000302
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 9%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(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] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 9%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(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] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The Gutman (or modified Schultz) index of a connected graph. This is ∑{u,v}⊆Vd(u)d(v)d(u,v) where d(u) is the degree of vertex u and d(u,v) is the distance between vertices u and v. For trees on n vertices, the modified Schultz index is related to the Wiener index via S∗(T)=4W(T)−(n−1)(2n−1) [1].
Matching statistic: St000467
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 9%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(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] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The hyper-Wiener index of a connected graph. This is ∑{u,v}⊆Vd(u,v)+d(u,v)2.
Matching statistic: St000771
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 9%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian (4−1−2−1−14−1−2−2−14−1−1−2−14). Its eigenvalues are 0,4,4,6, so the statistic is 2. The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000772
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 9%
Values
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 0 + 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 + 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian (4−1−2−1−14−1−2−2−14−1−1−2−14). Its eigenvalues are 0,4,4,6, so the statistic is 1. The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore also statistic 1. The graphs with statistic n−1, n−2 and n−3 have been characterised, see [1].
The following 16 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset.