Your data matches 29 different statistics following compositions of up to 3 maps.
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Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000503: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0
[2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
Description
The maximal difference between two elements in a common block.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[2,1] => [1,1,0,0]
=> [2] => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [4] => 4 = 3 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4 = 3 + 1
Description
The largest part of an integer composition.
Matching statistic: St000662
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
St000662: Permutations ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 3
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,3,4,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,4,6,2,5,7] => [1,5,3,6,2,4,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,4,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,5,2,4,6,7] => [1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,3,5,2,4,7,6] => [1,4,5,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,3,5,2,6,4,7] => [1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,5,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,3,5,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,3,5,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,5,6,2,4,7] => [1,5,6,4,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,5,6,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,6,2,4,5,7] => [1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,6,2,5,4,7] => [1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,6,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,6,5,2,4,7] => [1,5,6,4,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,3,6,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,4,2,5,6,3,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,2,6,3,5,7] => [1,5,3,6,2,4,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,2,6,5,3,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,3,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,4,3,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,4,3,5,6,2,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,3,6,2,5,7] => [1,5,3,6,2,4,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,3,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,5,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,4,5,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,4,5,2,6,3,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,5,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,4,5,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,4,5,3,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,5,6,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,6,2,5,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,6,3,2,5,7] => [1,5,6,4,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,6,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,6,5,2,3,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,4,6,5,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,5,2,3,6,4,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,5,2,4,3,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,5,2,4,3,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 3
[1,5,2,4,6,3,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,5,2,6,3,4,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
St000141: Permutations ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 3
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 2
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 2
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 1
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,5,7,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,2,3,6,5,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,7,4,5,6] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,7,4,6,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,7,5,4,6] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,7,6,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 2
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 2
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 2
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 3
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,5,7,3,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 3
[1,2,4,6,3,7,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 3
[1,2,4,6,5,7,3] => [1,2,7,5,4,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,6,7,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,7,3,5,6] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,7,3,6,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,7,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,7,6,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,4,7,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 2
[1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 3
[1,2,5,3,6,7,4] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
[1,2,5,3,7,4,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 4
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Mp00114: Permutations connectivity setBinary words
Mp00105: Binary words complementBinary words
St000392: Binary words ⟶ ℤResult quality: 81% values known / values provided: 81%distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => 0 => 0
[2,1] => 0 => 1 => 1
[1,2,3] => 11 => 00 => 0
[1,3,2] => 10 => 01 => 1
[2,1,3] => 01 => 10 => 1
[2,3,1] => 00 => 11 => 2
[3,1,2] => 00 => 11 => 2
[3,2,1] => 00 => 11 => 2
[1,2,3,4] => 111 => 000 => 0
[1,2,4,3] => 110 => 001 => 1
[1,3,2,4] => 101 => 010 => 1
[1,3,4,2] => 100 => 011 => 2
[1,4,2,3] => 100 => 011 => 2
[1,4,3,2] => 100 => 011 => 2
[2,1,3,4] => 011 => 100 => 1
[2,1,4,3] => 010 => 101 => 1
[2,3,1,4] => 001 => 110 => 2
[2,3,4,1] => 000 => 111 => 3
[2,4,1,3] => 000 => 111 => 3
[2,4,3,1] => 000 => 111 => 3
[3,1,2,4] => 001 => 110 => 2
[3,1,4,2] => 000 => 111 => 3
[3,2,1,4] => 001 => 110 => 2
[3,2,4,1] => 000 => 111 => 3
[3,4,1,2] => 000 => 111 => 3
[3,4,2,1] => 000 => 111 => 3
[4,1,2,3] => 000 => 111 => 3
[4,1,3,2] => 000 => 111 => 3
[4,2,1,3] => 000 => 111 => 3
[4,2,3,1] => 000 => 111 => 3
[4,3,1,2] => 000 => 111 => 3
[4,3,2,1] => 000 => 111 => 3
[1,2,3,4,5] => 1111 => 0000 => 0
[1,2,3,5,4] => 1110 => 0001 => 1
[1,2,4,3,5] => 1101 => 0010 => 1
[1,2,4,5,3] => 1100 => 0011 => 2
[1,2,5,3,4] => 1100 => 0011 => 2
[1,2,5,4,3] => 1100 => 0011 => 2
[1,3,2,4,5] => 1011 => 0100 => 1
[1,3,2,5,4] => 1010 => 0101 => 1
[1,3,4,2,5] => 1001 => 0110 => 2
[1,3,4,5,2] => 1000 => 0111 => 3
[1,3,5,2,4] => 1000 => 0111 => 3
[1,3,5,4,2] => 1000 => 0111 => 3
[1,4,2,3,5] => 1001 => 0110 => 2
[1,4,2,5,3] => 1000 => 0111 => 3
[1,4,3,2,5] => 1001 => 0110 => 2
[1,4,3,5,2] => 1000 => 0111 => 3
[1,4,5,2,3] => 1000 => 0111 => 3
[1,4,5,3,2] => 1000 => 0111 => 3
[8,6,7,5,4,3,2,1] => ? => ? => ? = 7
[7,6,8,4,5,3,2,1] => ? => ? => ? = 7
[8,6,5,4,7,3,2,1] => ? => ? => ? = 7
[8,5,6,4,7,3,2,1] => ? => ? => ? = 7
[8,6,4,5,7,3,2,1] => ? => ? => ? = 7
[8,5,4,6,7,3,2,1] => ? => ? => ? = 7
[8,4,5,6,7,3,2,1] => ? => ? => ? = 7
[8,7,6,4,3,5,2,1] => ? => ? => ? = 7
[7,8,6,4,3,5,2,1] => ? => ? => ? = 7
[8,6,7,4,3,5,2,1] => ? => ? => ? = 7
[7,8,3,4,5,6,2,1] => ? => ? => ? = 7
[8,5,4,6,3,7,2,1] => ? => ? => ? = 7
[8,6,5,7,4,2,3,1] => ? => ? => ? = 7
[8,7,5,4,6,2,3,1] => ? => ? => ? = 7
[8,7,4,5,6,2,3,1] => ? => ? => ? = 7
[8,6,5,4,7,2,3,1] => ? => ? => ? = 7
[8,5,6,4,7,2,3,1] => ? => ? => ? = 7
[8,6,4,5,7,2,3,1] => ? => ? => ? = 7
[8,4,5,6,7,2,3,1] => ? => ? => ? = 7
[8,7,6,5,3,2,4,1] => ? => ? => ? = 7
[8,6,7,5,3,2,4,1] => ? => ? => ? = 7
[8,7,5,6,3,2,4,1] => ? => ? => ? = 7
[7,8,5,6,2,3,4,1] => ? => ? => ? = 7
[8,7,6,4,3,2,5,1] => ? => ? => ? = 7
[7,8,6,4,3,2,5,1] => ? => ? => ? = 7
[8,6,7,4,3,2,5,1] => ? => ? => ? = 7
[8,7,6,3,4,2,5,1] => ? => ? => ? = 7
[8,6,7,3,4,2,5,1] => ? => ? => ? = 7
[7,6,8,3,4,2,5,1] => ? => ? => ? = 7
[7,6,8,4,2,3,5,1] => ? => ? => ? = 7
[7,8,6,3,2,4,5,1] => ? => ? => ? = 7
[8,7,5,4,3,2,6,1] => ? => ? => ? = 7
[8,7,5,3,4,2,6,1] => ? => ? => ? = 7
[7,8,5,3,4,2,6,1] => ? => ? => ? = 7
[8,7,5,3,2,4,6,1] => ? => ? => ? = 7
[7,8,3,4,2,5,6,1] => ? => ? => ? = 7
[7,8,4,2,3,5,6,1] => ? => ? => ? = 7
[8,6,5,3,4,2,7,1] => ? => ? => ? = 7
[8,5,6,3,4,2,7,1] => ? => ? => ? = 7
[8,4,3,5,6,2,7,1] => ? => ? => ? = 7
[8,5,6,4,2,3,7,1] => ? => ? => ? = 7
[8,6,2,3,4,5,7,1] => ? => ? => ? = 7
[8,3,4,5,2,6,7,1] => ? => ? => ? = 7
[8,5,4,2,3,6,7,1] => ? => ? => ? = 7
[8,5,2,3,4,6,7,1] => ? => ? => ? = 7
[8,4,2,3,5,6,7,1] => ? => ? => ? = 7
[7,5,6,4,3,2,8,1] => ? => ? => ? = 7
[7,4,5,6,3,2,8,1] => ? => ? => ? = 7
[7,6,5,3,4,2,8,1] => ? => ? => ? = 7
[7,5,6,3,4,2,8,1] => ? => ? => ? = 7
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000013
Mp00114: Permutations connectivity setBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000013: Dyck paths ⟶ ℤResult quality: 81% values known / values provided: 81%distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,5,3,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[8,6,7,5,4,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,8,4,5,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,4,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,4,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,4,5,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,4,6,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,5,6,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,4,3,5,2,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,6,4,3,5,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,4,3,5,2,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,3,4,5,6,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,4,6,3,7,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,7,4,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,4,6,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,4,5,6,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,4,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,4,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,4,5,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,5,6,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,5,3,2,4,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,5,3,2,4,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,6,3,2,4,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,5,6,2,3,4,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,4,3,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,6,4,3,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,4,3,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,3,4,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,3,4,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,8,3,4,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,8,4,2,3,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,6,3,2,4,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,4,3,2,6,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,3,4,2,6,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,5,3,4,2,6,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,3,2,4,6,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,3,4,2,5,6,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,4,2,3,5,6,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,3,4,2,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,3,4,2,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,3,5,6,2,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,4,2,3,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,2,3,4,5,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,3,4,5,2,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,4,2,3,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,2,3,4,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,2,3,5,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[7,5,6,4,3,2,8,1] => ? => ? => ?
=> ? = 7 + 1
[7,4,5,6,3,2,8,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,5,3,4,2,8,1] => ? => ? => ?
=> ? = 7 + 1
[7,5,6,3,4,2,8,1] => ? => ? => ?
=> ? = 7 + 1
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Mp00114: Permutations connectivity setBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 81% values known / values provided: 81%distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1,1] => [1,1]
=> 1 = 0 + 1
[2,1] => 0 => [2] => [2]
=> 2 = 1 + 1
[1,2,3] => 11 => [1,1,1] => [1,1,1]
=> 1 = 0 + 1
[1,3,2] => 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => 01 => [2,1] => [2,1]
=> 2 = 1 + 1
[2,3,1] => 00 => [3] => [3]
=> 3 = 2 + 1
[3,1,2] => 00 => [3] => [3]
=> 3 = 2 + 1
[3,2,1] => 00 => [3] => [3]
=> 3 = 2 + 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,4,3] => 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => [2,1,1]
=> 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => [3,1]
=> 3 = 2 + 1
[1,4,2,3] => 100 => [1,3] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => 100 => [1,3] => [3,1]
=> 3 = 2 + 1
[2,1,3,4] => 011 => [2,1,1] => [2,1,1]
=> 2 = 1 + 1
[2,1,4,3] => 010 => [2,2] => [2,2]
=> 2 = 1 + 1
[2,3,1,4] => 001 => [3,1] => [3,1]
=> 3 = 2 + 1
[2,3,4,1] => 000 => [4] => [4]
=> 4 = 3 + 1
[2,4,1,3] => 000 => [4] => [4]
=> 4 = 3 + 1
[2,4,3,1] => 000 => [4] => [4]
=> 4 = 3 + 1
[3,1,2,4] => 001 => [3,1] => [3,1]
=> 3 = 2 + 1
[3,1,4,2] => 000 => [4] => [4]
=> 4 = 3 + 1
[3,2,1,4] => 001 => [3,1] => [3,1]
=> 3 = 2 + 1
[3,2,4,1] => 000 => [4] => [4]
=> 4 = 3 + 1
[3,4,1,2] => 000 => [4] => [4]
=> 4 = 3 + 1
[3,4,2,1] => 000 => [4] => [4]
=> 4 = 3 + 1
[4,1,2,3] => 000 => [4] => [4]
=> 4 = 3 + 1
[4,1,3,2] => 000 => [4] => [4]
=> 4 = 3 + 1
[4,2,1,3] => 000 => [4] => [4]
=> 4 = 3 + 1
[4,2,3,1] => 000 => [4] => [4]
=> 4 = 3 + 1
[4,3,1,2] => 000 => [4] => [4]
=> 4 = 3 + 1
[4,3,2,1] => 000 => [4] => [4]
=> 4 = 3 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [2,1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [2,1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => [2,2,1]
=> 2 = 1 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[1,3,5,2,4] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[1,3,5,4,2] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[1,4,5,2,3] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[1,4,5,3,2] => 1000 => [1,4] => [4,1]
=> 4 = 3 + 1
[8,6,7,5,4,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,8,4,5,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,4,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,4,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,4,5,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,4,6,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,5,6,7,3,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,4,3,5,2,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,6,4,3,5,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,4,3,5,2,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,3,4,5,6,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,4,6,3,7,2,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,7,4,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,4,6,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,4,5,6,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,4,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,4,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,4,5,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,5,6,7,2,3,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,5,3,2,4,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,5,3,2,4,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,6,3,2,4,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,5,6,2,3,4,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,4,3,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,6,4,3,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,4,3,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,6,3,4,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,7,3,4,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,8,3,4,2,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,8,4,2,3,5,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,6,3,2,4,5,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,4,3,2,6,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,3,4,2,6,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,5,3,4,2,6,1] => ? => ? => ?
=> ? = 7 + 1
[8,7,5,3,2,4,6,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,3,4,2,5,6,1] => ? => ? => ?
=> ? = 7 + 1
[7,8,4,2,3,5,6,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,5,3,4,2,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,3,4,2,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,3,5,6,2,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,6,4,2,3,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,6,2,3,4,5,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,3,4,5,2,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,4,2,3,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,5,2,3,4,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[8,4,2,3,5,6,7,1] => ? => ? => ?
=> ? = 7 + 1
[7,5,6,4,3,2,8,1] => ? => ? => ?
=> ? = 7 + 1
[7,4,5,6,3,2,8,1] => ? => ? => ?
=> ? = 7 + 1
[7,6,5,3,4,2,8,1] => ? => ? => ?
=> ? = 7 + 1
[7,5,6,3,4,2,8,1] => ? => ? => ?
=> ? = 7 + 1
Description
The largest part of an integer partition.
Matching statistic: St001644
Mp00159: Permutations Demazure product with inversePermutations
Mp00160: Permutations graph of inversionsGraphs
Mp00117: Graphs Ore closureGraphs
St001644: Graphs ⟶ ℤResult quality: 76% values known / values provided: 76%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => ([],2)
=> ([],2)
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> ([],3)
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,1,2,4,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,3,4,5,6,2] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,4,2,5,6,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,4,3,5,6,2] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,5,2,3,6,4] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,5,2,4,6,3] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,6,2,3,4,5] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,6,2,3,5,4] => [1,6,3,4,5,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[2,3,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,3,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,3,4,6,5,1] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,3,5,4,6,1] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,4,1,5,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,4,1,6,3,5] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 5
[2,4,3,5,6,1] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,5,1,4,6,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,6,1,3,4,5] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[2,6,1,3,5,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,1,4,5,2,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[3,1,4,5,6,2] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,1,4,6,5,2] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,1,5,4,6,2] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,2,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[3,2,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,2,4,6,1,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,2,4,6,5,1] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,2,5,4,6,1] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,4,1,5,6,2] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[3,4,2,5,6,1] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,2,5,3,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[4,1,2,5,6,3] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,2,6,3,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,2,6,5,3] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,3,5,2,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[4,1,3,5,6,2] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,3,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,3,6,5,2] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,5,2,6,3] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,1,5,3,6,2] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,2,1,5,6,3] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[4,2,3,5,6,1] => [6,3,2,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
Description
The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Mp00159: Permutations Demazure product with inversePermutations
Mp00159: Permutations Demazure product with inversePermutations
Mp00160: Permutations graph of inversionsGraphs
St001330: Graphs ⟶ ℤResult quality: 74% values known / values provided: 74%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,4,5,6,1] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,3,4,6,1,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,3,4,6,5,1] => [6,2,3,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,3,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[2,3,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[2,3,6,1,5,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[2,4,1,5,6,3] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 5 + 1
[2,4,1,6,5,3] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,4,3,5,6,1] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,4,3,6,1,5] => [5,3,2,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,4,3,6,5,1] => [6,3,2,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,5,1,3,6,4] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,5,1,4,6,3] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,5,1,6,3,4] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,5,1,6,4,3] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,6,1,3,4,5] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,6,1,3,5,4] => [3,6,1,4,5,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,6,1,4,3,5] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,6,1,4,5,3] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,6,1,5,3,4] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[2,6,1,5,4,3] => [3,6,1,5,4,2] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,1,4,5,6,2] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,1,4,6,2,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,1,4,6,5,2] => [6,2,3,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[3,1,6,2,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[3,1,6,2,5,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[3,2,4,5,6,1] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,2,4,6,1,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,2,4,6,5,1] => [6,2,3,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[3,2,6,1,4,5] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[3,2,6,1,5,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 + 1
[3,4,1,5,6,2] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,4,1,6,2,5] => [5,3,2,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,4,1,6,5,2] => [6,3,2,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,4,2,5,6,1] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,4,2,6,1,5] => [5,3,2,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[3,4,2,6,5,1] => [6,3,2,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,1,2,5,6,3] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,1,2,6,3,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,1,2,6,5,3] => [6,2,3,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,1,3,5,6,2] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,1,3,6,2,5] => [5,2,3,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,1,3,6,5,2] => [6,2,3,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,2,1,5,6,3] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,2,1,6,3,5] => [5,3,2,6,1,4] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,2,1,6,5,3] => [6,3,2,5,4,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
[4,2,3,5,6,1] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => ([(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)
=> ? = 5 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000010
Mp00160: Permutations graph of inversionsGraphs
Mp00037: Graphs to partition of connected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 49% values known / values provided: 49%distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
=> [1,1]
=> [2]
=> 1 = 0 + 1
[2,1] => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2 = 1 + 1
[1,2,3] => ([],3)
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,3] => ([(1,2)],3)
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[1,2,3,4] => ([],4)
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,2,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => ([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,4,5] => ([],5)
=> [1,1,1,1,1]
=> [5]
=> 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,3,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,3,2,4,5] => ([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 2 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4 = 3 + 1
[7,8,6,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,6,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,4,5,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,4,6,7,3,2,1] => ?
=> ?
=> ?
=> ? = 7 + 1
[8,7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,6,7,3,4,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,5,4,3,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,5,3,4,6,2,1] => ?
=> ?
=> ?
=> ? = 7 + 1
[7,8,5,3,4,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,4,3,5,6,2,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,4,6,3,7,2,1] => ?
=> ?
=> ?
=> ? = 7 + 1
[8,7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,5,4,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,5,7,4,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,6,7,4,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,4,5,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,5,4,7,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,6,4,7,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,4,5,7,2,3,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,5,6,7,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,7,5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,5,7,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,6,4,3,2,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,4,3,2,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,8,6,3,4,2,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[8,6,7,3,4,2,5,1] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
[7,6,8,3,4,2,5,1] => ([(0,3),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ?
=> ? = 7 + 1
Description
The length of the partition.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St001120The length of a longest path in a graph. St001268The size of the largest ordinal summand in the poset. St000171The degree of the graph. St001645The pebbling number of a connected graph. St000209Maximum difference of elements in cycles. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.