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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000253
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(load all 4 compositions to match this statistic)
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 1
{{1},{2}}
=> 0
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 1
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 1
{{1,2,3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> 1
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 2
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 2
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000254
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(load all 2 compositions to match this statistic)
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
St000254: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St000254: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> {{1,2}}
=> 1
{{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> 1
{{1,2},{3}}
=> {{1,2},{3}}
=> 1
{{1,3},{2}}
=> {{1,3},{2}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 2
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 1
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> 1
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 2
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> 2
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 2
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> 2
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 1
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 2
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 1
{{1,2,5,6,7,8},{3,4}}
=> {{1,2,4},{3,5,6,7,8}}
=> ? = 1
{{1,2,3,6,7,8},{4,5}}
=> {{1,2,3,5},{4,6,7,8}}
=> ? = 1
{{1,2,3,4,7,8},{5,6}}
=> {{1,2,3,4,6},{5,7,8}}
=> ? = 1
{{1,8},{2,3,4,5,6,7}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 1
{{1,3,5,6,7,8},{2,4}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
{{1,3,4,6,7,8},{2,5}}
=> {{1,4,5},{2,3,6,7,8}}
=> ? = 2
{{1,2,4,6,7,8},{3,5}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 2
{{1,3,4,5,7,8},{2,6}}
=> {{1,4,7,8},{2,3,5,6}}
=> ? = 2
{{1,2,4,5,7,8},{3,6}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 2
{{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
{{1,3,4,5,6,8},{2,7}}
=> {{1,4,6,7},{2,3,5,8}}
=> ? = 2
{{1,2,4,5,6,8},{3,7}}
=> {{1,2,5,8},{3,4,6,7}}
=> ? = 2
{{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 2
{{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
{{1,3,4,5,6,7},{2,8}}
=> {{1,4,6,8},{2,3,5,7}}
=> ? = 2
{{1,2,4,5,6,7},{3,8}}
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 2
{{1,2,3,5,6,7},{4,8}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 2
{{1,2,3,4,6,7},{5,8}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2
{{1,4},{2,3,5,6,7,8}}
=> {{1,3,4},{2,5,6,7,8}}
=> ? = 2
{{1,5},{2,3,4,6,7,8}}
=> {{1,3,6,7,8},{2,4,5}}
=> ? = 2
{{1,6},{2,3,4,5,7,8}}
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 2
{{1,7},{2,3,4,5,6,8}}
=> {{1,3,5,8},{2,4,6,7}}
=> ? = 2
Description
The nesting number of a set partition.
This is the maximal number of chords in the standard representation of a set partition that mutually nest.
Matching statistic: St000451
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,2,1] => 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,3,1] => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 2 = 1 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 2 = 1 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 2 = 1 + 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => [6,7,5,4,3,2,1] => ? = 1 + 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => [5,7,6,4,3,2,1] => ? = 1 + 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [6,4,3,2,1,7,5] => ? = 2 + 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [6,5,7,4,3,2,1] => ? = 1 + 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => [4,3,2,1,7,6,5] => ? = 1 + 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => [4,3,2,1,6,5,7] => ? = 1 + 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [5,6,7,4,3,2,1] => ? = 1 + 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => [4,3,2,1,6,7,5] => ? = 1 + 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [4,3,2,1,5,7,6] => ? = 1 + 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => [4,7,6,5,3,2,1] => ? = 1 + 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [6,1,2,7,3,5,4] => [6,5,3,2,1,7,4] => ? = 2 + 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => ? = 2 + 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [5,1,2,7,3,4,6] => [5,3,2,1,7,6,4] => ? = 2 + 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [5,1,2,6,3,4,7] => [5,3,2,1,6,4,7] => ? = 2 + 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => [4,6,7,5,3,2,1] => ? = 1 + 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [5,1,2,7,3,6,4] => [5,3,2,1,6,7,4] => ? = 2 + 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [5,1,2,4,3,7,6] => [4,5,3,2,1,7,6] => ? = 1 + 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => [4,5,3,2,1,6,7] => ? = 1 + 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => [5,4,7,6,3,2,1] => ? = 1 + 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [6,1,2,7,4,3,5] => [6,3,2,1,7,5,4] => ? = 2 + 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => [6,5,4,7,3,2,1] => ? = 1 + 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => [3,2,1,7,6,5,4] => ? = 1 + 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => [3,2,1,6,5,4,7] => ? = 1 + 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => [5,4,6,7,3,2,1] => ? = 1 + 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [3,1,2,7,4,6,5] => [3,2,1,6,7,5,4] => ? = 1 + 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 1 + 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => [4,5,7,6,3,2,1] => ? = 1 + 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [6,1,2,7,5,3,4] => [5,6,3,2,1,7,4] => ? = 2 + 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => [4,6,3,2,1,7,5] => ? = 2 + 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [5,6,4,7,3,2,1] => ? = 1 + 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [3,1,2,7,5,4,6] => [3,2,1,5,7,6,4] => ? = 1 + 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [3,1,2,6,7,4,5] => [3,2,1,6,4,7,5] => ? = 2 + 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [3,1,2,6,5,4,7] => [3,2,1,5,6,4,7] => ? = 1 + 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => [4,6,5,7,3,2,1] => ? = 1 + 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [3,1,2,7,6,5,4] => [3,2,1,6,5,7,4] => ? = 1 + 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [3,2,1,4,7,6,5] => ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 1 + 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => [4,5,6,7,3,2,1] => ? = 1 + 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [3,1,2,7,5,6,4] => [3,2,1,5,6,7,4] => ? = 1 + 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [3,1,2,4,7,6,5] => [3,2,1,4,6,7,5] => ? = 1 + 1
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 1 + 1
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [7,1,3,2,4,5,6] => [3,7,6,5,4,2,1] => ? = 1 + 1
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [6,1,7,2,4,5,3] => [6,5,4,2,1,7,3] => ? = 2 + 1
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [7,1,6,2,4,3,5] => [6,3,7,5,4,2,1] => ? = 2 + 1
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [5,1,7,2,4,3,6] => [5,4,2,1,7,6,3] => ? = 2 + 1
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [5,1,6,2,4,3,7] => [5,4,2,1,6,3,7] => ? = 2 + 1
{{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [7,1,3,2,4,6,5] => [3,6,7,5,4,2,1] => ? = 1 + 1
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [5,1,7,2,4,6,3] => [5,4,2,1,6,7,3] => ? = 2 + 1
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [5,1,3,2,4,7,6] => [3,5,4,2,1,7,6] => ? = 1 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000455
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
{{1,2,3}}
=> [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
{{1,2,3,4}}
=> [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
{{1,2,3,4,5}}
=> [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
{{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,4},{2,3,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,5},{2,3,4}}
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,3,4,5}}
=> [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,3,4},{5}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1,5},{2,3},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,3,5},{4}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,3},{4,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 - 1
{{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,4},{2,5},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,4},{2},{3,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,4,5},{3}}
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,4},{3,5}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 - 1
{{1,5},{2},{3,4}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2,5},{3,4}}
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2},{3,4,5}}
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 - 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> 0 = 1 - 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 - 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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 - 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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
{{1,2,3,4,5,6}}
=> [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,4,5},{6}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,4,6},{5}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,4},{5,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,3,4},{5},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,5,6},{4}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,5},{4,6}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
{{1,2,3,5},{4},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3,6},{4,5}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,3},{4,5,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,3,6},{4},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4,5,6},{3}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4,5},{3},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4,6},{3},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,5,6},{3},{4}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4,5,6},{2}}
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4,5},{2},{6}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4,6},{2},{5}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => ([(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)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
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