Your data matches 32 different statistics following compositions of up to 3 maps.
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Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St000428: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 10
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 6
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 10
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 7
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 11
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 10
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 9
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 6
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 6
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 13
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 11
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => 10
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 12
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 9
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 8
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 7
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => 6
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 7
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 6
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => 4
Description
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00069: Permutations complementPermutations
St000423: Permutations ⟶ ℤResult quality: 69% values known / values provided: 69%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,3,4,2] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 10
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,4,5,3,2] => 6
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => 4
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,3,2,4] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [2,1,3,4] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,4,2,1,5] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [5,4,3,2,1,6] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [1,5,6,4,3,2] => 10
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,5,4,2] => 7
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,3,5,2] => 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,5,4,3] => 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,1,5] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,2,4,1,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [4,5,3,2,1,6] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => 11
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [1,5,4,6,3,2] => 10
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 9
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,3,4,5,2] => 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,3,2,5] => 6
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [2,1,4,5,3] => 6
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [2,3,1,5,4] => 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [2,3,4,1,5] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [3,5,4,2,1,6] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,2,1,4,5] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [4,3,5,2,1,6] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => 13
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [1,4,5,6,3,2] => 11
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [1,5,4,3,6,2] => 10
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 12
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,4,5,3] => 9
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,3,2,5,4] => 8
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,3,4,2,5] => 7
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [2,5,4,3,1,6] => 6
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [2,1,3,5,4] => 7
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [2,1,4,3,5] => 6
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,3,1,4,5] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [3,4,5,2,1,6] => 4
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => ? = 4
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => [4,3,5,6,2,1,7] => ? = 7
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [4,5,3,2,6,1,7] => ? = 6
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [5,4,3,2,1,6,7] => ? = 5
[5,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => [2,5,4,6,3,1,7] => ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,7,1] => [3,4,5,6,2,1,7] => ? = 10
[4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [3,5,4,2,6,1,7] => ? = 9
[3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [4,3,2,6,5,1,7] => ? = 9
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [4,3,5,2,6,1,7] => ? = 8
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [4,5,3,2,1,6,7] => ? = 7
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,6,1] => [1,5,4,6,3,2,7] => ? = 17
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => [2,4,5,6,3,1,7] => ? = 14
[5,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [2,5,4,3,6,1,7] => ? = 13
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,7,1] => [3,2,5,6,4,1,7] => ? = 13
[4,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [3,4,2,6,5,1,7] => ? = 12
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [3,4,5,2,6,1,7] => ? = 11
[4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [3,5,4,2,1,6,7] => ? = 10
[3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [4,3,2,5,6,1,7] => ? = 10
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [4,3,5,2,1,6,7] => ? = 9
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => [1,5,4,3,6,2,7] => ? = 18
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,7,1] => [2,3,5,6,4,1,7] => ? = 17
[5,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [2,4,3,6,5,1,7] => ? = 16
[5,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [2,4,5,3,6,1,7] => ? = 15
[5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [2,5,4,3,1,6,7] => ? = 14
[4,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [3,2,4,6,5,1,7] => ? = 15
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [3,2,5,4,6,1,7] => ? = 14
[4,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [3,4,2,5,6,1,7] => ? = 13
[4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [3,4,5,2,1,6,7] => ? = 12
[3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => ? = 12
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [4,3,2,5,1,6,7] => ? = 11
[6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [1,5,4,3,2,6,7] => ? = 19
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [2,1,5,6,4,3,7] => ? = 21
[5,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => [2,3,4,6,5,1,7] => ? = 19
[5,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [2,3,5,4,6,1,7] => ? = 18
[5,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [2,4,3,5,6,1,7] => ? = 17
[5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [2,4,5,3,1,6,7] => ? = 16
[4,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 18
[4,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [3,2,4,5,6,1,7] => ? = 16
[4,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [3,2,5,4,1,6,7] => ? = 15
[4,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [3,4,2,1,6,5,7] => ? = 15
[4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [3,4,2,5,1,6,7] => ? = 14
[3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 13
[5,5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [6,7,4,2,3,5,1] => [2,1,4,6,5,3,7] => ? = 23
[5,5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [2,1,5,4,6,3,7] => ? = 22
[5,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => ? = 22
[5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [2,3,4,5,6,1,7] => ? = 20
[5,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 19
[5,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [2,4,3,1,6,5,7] => ? = 19
[5,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => [2,4,3,5,1,6,7] => ? = 18
[4,4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 19
Description
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
Matching statistic: St000185
Mp00044: Integer partitions conjugateInteger partitions
St000185: Integer partitions ⟶ ℤResult quality: 57% values known / values provided: 57%distinct values known / distinct values provided: 61%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 0
[3]
=> [1,1,1]
=> 3
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 0
[4]
=> [1,1,1,1]
=> 6
[3,1]
=> [2,1,1]
=> 3
[2,2]
=> [2,2]
=> 2
[2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [4]
=> 0
[5]
=> [1,1,1,1,1]
=> 10
[4,1]
=> [2,1,1,1]
=> 6
[3,2]
=> [2,2,1]
=> 4
[3,1,1]
=> [3,1,1]
=> 3
[2,2,1]
=> [3,2]
=> 2
[2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [5]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 10
[4,2]
=> [2,2,1,1]
=> 7
[4,1,1]
=> [3,1,1,1]
=> 6
[3,3]
=> [2,2,2]
=> 6
[3,2,1]
=> [3,2,1]
=> 4
[3,1,1,1]
=> [4,1,1]
=> 3
[2,2,2]
=> [3,3]
=> 3
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 11
[5,1,1]
=> [3,1,1,1,1]
=> 10
[4,3]
=> [2,2,2,1]
=> 9
[4,2,1]
=> [3,2,1,1]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> 6
[3,3,1]
=> [3,2,2]
=> 6
[3,2,2]
=> [3,3,1]
=> 5
[3,2,1,1]
=> [4,2,1]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> 3
[2,2,2,1]
=> [4,3]
=> 3
[2,2,1,1,1]
=> [5,2]
=> 2
[5,3]
=> [2,2,2,1,1]
=> 13
[5,2,1]
=> [3,2,1,1,1]
=> 11
[5,1,1,1]
=> [4,1,1,1,1]
=> 10
[4,4]
=> [2,2,2,2]
=> 12
[4,3,1]
=> [3,2,2,1]
=> 9
[4,2,2]
=> [3,3,1,1]
=> 8
[4,2,1,1]
=> [4,2,1,1]
=> 7
[4,1,1,1,1]
=> [5,1,1,1]
=> 6
[3,3,2]
=> [3,3,2]
=> 7
[3,3,1,1]
=> [4,2,2]
=> 6
[3,2,2,1]
=> [4,3,1]
=> 5
[3,2,1,1,1]
=> [5,2,1]
=> 4
[3,3,2,1,1,1]
=> [6,3,2]
=> ? = 7
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> ? = 12
[4,3,2,1,1,1]
=> [6,3,2,1]
=> ? = 10
[4,2,2,2,1,1]
=> [6,4,1,1]
=> ? = 9
[3,3,3,1,1,1]
=> [6,3,3]
=> ? = 9
[3,2,2,2,2,1]
=> [6,5,1]
=> ? = 7
[6,2,2,1,1,1]
=> [6,3,1,1,1,1]
=> ? = 17
[5,3,2,1,1,1]
=> [6,3,2,1,1]
=> ? = 14
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> ? = 13
[4,4,2,1,1,1]
=> [6,3,2,2]
=> ? = 13
[4,3,3,1,1,1]
=> [6,3,3,1]
=> ? = 12
[4,3,2,2,1,1]
=> [6,4,2,1]
=> ? = 11
[4,2,2,2,2,1]
=> [6,5,1,1]
=> ? = 10
[3,3,3,2,1,1]
=> [6,4,3]
=> ? = 10
[3,3,2,2,2,1]
=> [6,5,2]
=> ? = 9
[6,3,2,1,1,1]
=> [6,3,2,1,1,1]
=> ? = 19
[6,2,2,2,1,1]
=> [6,4,1,1,1,1]
=> ? = 18
[5,4,2,1,1,1]
=> [6,3,2,2,1]
=> ? = 17
[5,3,3,1,1,1]
=> [6,3,3,1,1]
=> ? = 16
[5,3,2,2,1,1]
=> [6,4,2,1,1]
=> ? = 15
[5,2,2,2,2,1]
=> [6,5,1,1,1]
=> ? = 14
[4,4,3,1,1,1]
=> [6,3,3,2]
=> ? = 15
[4,4,2,2,1,1]
=> [6,4,2,2]
=> ? = 14
[4,3,3,2,1,1]
=> [6,4,3,1]
=> ? = 13
[4,3,2,2,2,1]
=> [6,5,2,1]
=> ? = 12
[3,3,3,3,1,1]
=> [6,4,4]
=> ? = 12
[3,3,3,2,2,1]
=> [6,5,3]
=> ? = 11
[6,4,2,1,1,1]
=> [6,3,2,2,1,1]
=> ? = 22
[6,3,3,1,1,1]
=> [6,3,3,1,1,1]
=> ? = 21
[6,3,2,2,1,1]
=> [6,4,2,1,1,1]
=> ? = 20
[6,2,2,2,2,1]
=> [6,5,1,1,1,1]
=> ? = 19
[5,5,2,1,1,1]
=> [6,3,2,2,2]
=> ? = 21
[5,4,3,1,1,1]
=> [6,3,3,2,1]
=> ? = 19
[5,4,2,2,1,1]
=> [6,4,2,2,1]
=> ? = 18
[5,3,3,2,1,1]
=> [6,4,3,1,1]
=> ? = 17
[5,3,2,2,2,1]
=> [6,5,2,1,1]
=> ? = 16
[4,4,4,1,1,1]
=> [6,3,3,3]
=> ? = 18
[4,4,3,2,1,1]
=> [6,4,3,2]
=> ? = 16
[4,4,2,2,2,1]
=> [6,5,2,2]
=> ? = 15
[4,3,3,3,1,1]
=> [6,4,4,1]
=> ? = 15
[4,3,3,2,2,1]
=> [6,5,3,1]
=> ? = 14
[3,3,3,3,2,1]
=> [6,5,4]
=> ? = 13
[6,5,2,1,1,1]
=> [6,3,2,2,2,1]
=> ? = 26
[6,4,3,1,1,1]
=> [6,3,3,2,1,1]
=> ? = 24
[6,4,2,2,1,1]
=> [6,4,2,2,1,1]
=> ? = 23
[6,3,3,2,1,1]
=> [6,4,3,1,1,1]
=> ? = 22
[6,3,2,2,2,1]
=> [6,5,2,1,1,1]
=> ? = 21
[5,5,3,1,1,1]
=> [6,3,3,2,2]
=> ? = 23
[5,5,2,2,1,1]
=> [6,4,2,2,2]
=> ? = 22
[5,4,4,1,1,1]
=> [6,3,3,3,1]
=> ? = 22
Description
The weighted size of a partition. Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is $$\sum_{i=0}^m i \cdot \lambda_i.$$ This is also the sum of the leg lengths of the cells in $\lambda$, or $$ \sum_i \binom{\lambda^{\prime}_i}{2} $$ where $\lambda^{\prime}$ is the conjugate partition of $\lambda$. This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2]. This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].
Matching statistic: St000169
Mp00044: Integer partitions conjugateInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 58%
Values
[1]
=> [1]
=> [[1]]
=> 0
[2]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1]
=> [2]
=> [[1,2]]
=> 0
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[2,1]
=> [2,1]
=> [[1,2],[3]]
=> 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 0
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 6
[3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 10
[4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 6
[3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 4
[3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
[2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 10
[4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 6
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 6
[3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 4
[3,1,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[2,2,1,1]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 11
[5,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 10
[4,3]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 6
[3,3,1]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 6
[3,2,2]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 5
[3,2,1,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
[2,2,2,1]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 2
[5,3]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 13
[5,2,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> 11
[5,1,1,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 10
[4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 12
[4,3,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 9
[4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 8
[4,2,1,1]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> 7
[4,1,1,1,1]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> 6
[3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 7
[3,3,1,1]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 6
[3,2,2,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 5
[3,2,1,1,1]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> 4
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 7
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 6
[2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 5
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11],[12]]
=> ? = 12
[4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12]]
=> ? = 10
[4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11],[12]]
=> ? = 9
[3,3,3,1,1,1]
=> [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? = 9
[3,2,2,2,2,1]
=> [6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? = 7
[6,2,2,1,1,1]
=> [6,3,1,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11],[12],[13]]
=> ? = 17
[5,3,2,1,1,1]
=> [6,3,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12],[13]]
=> ? = 14
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11],[12],[13]]
=> ? = 13
[4,4,2,1,1,1]
=> [6,3,2,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12,13]]
=> ? = 13
[4,3,3,1,1,1]
=> [6,3,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13]]
=> ? = 12
[4,3,2,2,1,1]
=> [6,4,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? = 11
[4,2,2,2,2,1]
=> [6,5,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12],[13]]
=> ? = 10
[3,3,3,2,1,1]
=> [6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 10
[3,3,2,2,2,1]
=> [6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? = 9
[6,3,2,1,1,1]
=> [6,3,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12],[13],[14]]
=> ? = 19
[6,2,2,2,1,1]
=> [6,4,1,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11],[12],[13],[14]]
=> ? = 18
[5,4,2,1,1,1]
=> [6,3,2,2,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12,13],[14]]
=> ? = 17
[5,3,3,1,1,1]
=> [6,3,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13],[14]]
=> ? = 16
[5,3,2,2,1,1]
=> [6,4,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13],[14]]
=> ? = 15
[5,2,2,2,2,1]
=> [6,5,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12],[13],[14]]
=> ? = 14
[4,4,3,1,1,1]
=> [6,3,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? = 15
[4,4,2,2,1,1]
=> [6,4,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13,14]]
=> ? = 14
[4,3,3,2,1,1]
=> [6,4,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14]]
=> ? = 13
[4,3,2,2,2,1]
=> [6,5,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14]]
=> ? = 12
[3,3,3,3,1,1]
=> [6,4,4]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13,14]]
=> ? = 12
[3,3,3,2,2,1]
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 11
[6,4,2,1,1,1]
=> [6,3,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12,13],[14],[15]]
=> ? = 22
[6,3,3,1,1,1]
=> [6,3,3,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13],[14],[15]]
=> ? = 21
[6,3,2,2,1,1]
=> [6,4,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13],[14],[15]]
=> ? = 20
[6,2,2,2,2,1]
=> [6,5,1,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12],[13],[14],[15]]
=> ? = 19
[5,5,2,1,1,1]
=> [6,3,2,2,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12,13],[14,15]]
=> ? = 21
[5,4,3,1,1,1]
=> [6,3,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14],[15]]
=> ? = 19
[5,4,2,2,1,1]
=> [6,4,2,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13,14],[15]]
=> ? = 18
[5,3,3,2,1,1]
=> [6,4,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14],[15]]
=> ? = 17
[5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14],[15]]
=> ? = 16
[4,4,4,1,1,1]
=> [6,3,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 18
[4,4,3,2,1,1]
=> [6,4,3,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14,15]]
=> ? = 16
[4,4,2,2,2,1]
=> [6,5,2,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14,15]]
=> ? = 15
[4,3,3,3,1,1]
=> [6,4,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13,14],[15]]
=> ? = 15
[4,3,3,2,2,1]
=> [6,5,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15]]
=> ? = 14
[3,3,3,3,2,1]
=> [6,5,4]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15]]
=> ? = 13
[6,5,2,1,1,1]
=> [6,3,2,2,2,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11],[12,13],[14,15],[16]]
=> ? = 26
[6,4,3,1,1,1]
=> [6,3,3,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14],[15],[16]]
=> ? = 24
[6,4,2,2,1,1]
=> [6,4,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 23
[6,3,3,2,1,1]
=> [6,4,3,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14],[15],[16]]
=> ? = 22
[6,3,2,2,2,1]
=> [6,5,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14],[15],[16]]
=> ? = 21
[5,5,3,1,1,1]
=> [6,3,3,2,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14],[15,16]]
=> ? = 23
Description
The cocharge of a standard tableau. The '''cocharge''' of a standard tableau $T$, denoted $\mathrm{cc}(T)$, is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation $w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from $w_n$, scan the entries right-to-left until finding the entry $1$ with a superscript $0$. 2) Continue scanning until the $2$ is found, and label this with a superscript $1$. Then scan until the $3$ is found, labeling with a $2$, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.
Matching statistic: St000330
Mp00044: Integer partitions conjugateInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 58%
Values
[1]
=> [1]
=> [[1]]
=> 0
[2]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1]
=> [2]
=> [[1,2]]
=> 0
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 0
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 6
[3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 10
[4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 6
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4
[3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 10
[4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 6
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 6
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 3
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 11
[5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 10
[4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 6
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 6
[3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 5
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 3
[2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 2
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 13
[5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 11
[5,1,1,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> 10
[4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 12
[4,3,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 9
[4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 8
[4,2,1,1]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 7
[4,1,1,1,1]
=> [5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> 6
[3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 7
[3,3,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 6
[3,2,2,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 5
[3,2,1,1,1]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 4
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 7
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 6
[2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> [[1,5,6,10,11,12],[2,8,9],[3],[4],[7]]
=> ? = 12
[4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 10
[4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 9
[3,3,3,1,1,1]
=> [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 9
[3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 7
[6,2,2,1,1,1]
=> [6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? = 17
[5,3,2,1,1,1]
=> [6,3,2,1,1]
=> [[1,4,7,11,12,13],[2,6,10],[3,9],[5],[8]]
=> ? = 14
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> [[1,5,6,7,12,13],[2,9,10,11],[3],[4],[8]]
=> ? = 13
[4,4,2,1,1,1]
=> [6,3,2,2]
=> [[1,2,7,11,12,13],[3,4,10],[5,6],[8,9]]
=> ? = 13
[4,3,3,1,1,1]
=> [6,3,3,1]
=> [[1,3,4,11,12,13],[2,6,7],[5,9,10],[8]]
=> ? = 12
[4,3,2,2,1,1]
=> [6,4,2,1]
=> [[1,3,6,7,12,13],[2,5,10,11],[4,9],[8]]
=> ? = 11
[4,2,2,2,2,1]
=> [6,5,1,1]
=> [[1,4,5,6,7,13],[2,9,10,11,12],[3],[8]]
=> ? = 10
[3,3,3,2,1,1]
=> [6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? = 10
[3,3,2,2,2,1]
=> [6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? = 9
[6,3,2,1,1,1]
=> [6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 19
[6,2,2,2,1,1]
=> [6,4,1,1,1,1]
=> [[1,6,7,8,13,14],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 18
[5,4,2,1,1,1]
=> [6,3,2,2,1]
=> [[1,3,8,12,13,14],[2,5,11],[4,7],[6,10],[9]]
=> ? = 17
[5,3,3,1,1,1]
=> [6,3,3,1,1]
=> [[1,4,5,12,13,14],[2,7,8],[3,10,11],[6],[9]]
=> ? = 16
[5,3,2,2,1,1]
=> [6,4,2,1,1]
=> [[1,4,7,8,13,14],[2,6,11,12],[3,10],[5],[9]]
=> ? = 15
[5,2,2,2,2,1]
=> [6,5,1,1,1]
=> [[1,5,6,7,8,14],[2,10,11,12,13],[3],[4],[9]]
=> ? = 14
[4,4,3,1,1,1]
=> [6,3,3,2]
=> [[1,2,5,12,13,14],[3,4,8],[6,7,11],[9,10]]
=> ? = 15
[4,4,2,2,1,1]
=> [6,4,2,2]
=> [[1,2,7,8,13,14],[3,4,11,12],[5,6],[9,10]]
=> ? = 14
[4,3,3,2,1,1]
=> [6,4,3,1]
=> [[1,3,4,8,13,14],[2,6,7,12],[5,10,11],[9]]
=> ? = 13
[4,3,2,2,2,1]
=> [6,5,2,1]
=> [[1,3,6,7,8,14],[2,5,11,12,13],[4,10],[9]]
=> ? = 12
[3,3,3,3,1,1]
=> [6,4,4]
=> [[1,2,3,4,13,14],[5,6,7,8],[9,10,11,12]]
=> ? = 12
[3,3,3,2,2,1]
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? = 11
[6,4,2,1,1,1]
=> [6,3,2,2,1,1]
=> [[1,4,9,13,14,15],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? = 22
[6,3,3,1,1,1]
=> [6,3,3,1,1,1]
=> [[1,5,6,13,14,15],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 21
[6,3,2,2,1,1]
=> [6,4,2,1,1,1]
=> [[1,5,8,9,14,15],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? = 20
[6,2,2,2,2,1]
=> [6,5,1,1,1,1]
=> [[1,6,7,8,9,15],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? = 19
[5,5,2,1,1,1]
=> [6,3,2,2,2]
=> [[1,2,9,13,14,15],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 21
[5,4,3,1,1,1]
=> [6,3,3,2,1]
=> [[1,3,6,13,14,15],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 19
[5,4,2,2,1,1]
=> [6,4,2,2,1]
=> [[1,3,8,9,14,15],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 18
[5,3,3,2,1,1]
=> [6,4,3,1,1]
=> [[1,4,5,9,14,15],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 17
[5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [[1,4,7,8,9,15],[2,6,12,13,14],[3,11],[5],[10]]
=> ? = 16
[4,4,4,1,1,1]
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 18
[4,4,3,2,1,1]
=> [6,4,3,2]
=> [[1,2,5,9,14,15],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 16
[4,4,2,2,2,1]
=> [6,5,2,2]
=> [[1,2,7,8,9,15],[3,4,12,13,14],[5,6],[10,11]]
=> ? = 15
[4,3,3,3,1,1]
=> [6,4,4,1]
=> [[1,3,4,5,14,15],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 15
[4,3,3,2,2,1]
=> [6,5,3,1]
=> [[1,3,4,8,9,15],[2,6,7,13,14],[5,11,12],[10]]
=> ? = 14
[3,3,3,3,2,1]
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> ? = 13
[6,5,2,1,1,1]
=> [6,3,2,2,2,1]
=> [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? = 26
[6,4,3,1,1,1]
=> [6,3,3,2,1,1]
=> [[1,4,7,14,15,16],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? = 24
[6,4,2,2,1,1]
=> [6,4,2,2,1,1]
=> [[1,4,9,10,15,16],[2,6,13,14],[3,8],[5,12],[7],[11]]
=> ? = 23
[6,3,3,2,1,1]
=> [6,4,3,1,1,1]
=> [[1,5,6,10,15,16],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? = 22
[6,3,2,2,2,1]
=> [6,5,2,1,1,1]
=> [[1,5,8,9,10,16],[2,7,13,14,15],[3,12],[4],[6],[11]]
=> ? = 21
[5,5,3,1,1,1]
=> [6,3,3,2,2]
=> [[1,2,7,14,15,16],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? = 23
Description
The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Mp00044: Integer partitions conjugateInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
St000336: Standard tableaux ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 58%
Values
[1]
=> [1]
=> [[1]]
=> 0
[2]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1]
=> [2]
=> [[1,2]]
=> 0
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 0
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 6
[3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 10
[4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 6
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4
[3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 2
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 10
[4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 7
[4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 6
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 6
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 3
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 11
[5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 10
[4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 9
[4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 7
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 6
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 6
[3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 5
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 3
[2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 2
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 13
[5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 11
[5,1,1,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> 10
[4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 12
[4,3,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 9
[4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 8
[4,2,1,1]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 7
[4,1,1,1,1]
=> [5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> 6
[3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 7
[3,3,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 6
[3,2,2,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 5
[3,2,1,1,1]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 4
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 7
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 6
[2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 5
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> [[1,5,6,10,11,12],[2,8,9],[3],[4],[7]]
=> ? = 12
[4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 10
[4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 9
[3,3,3,1,1,1]
=> [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? = 9
[3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 7
[6,2,2,1,1,1]
=> [6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? = 17
[5,3,2,1,1,1]
=> [6,3,2,1,1]
=> [[1,4,7,11,12,13],[2,6,10],[3,9],[5],[8]]
=> ? = 14
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> [[1,5,6,7,12,13],[2,9,10,11],[3],[4],[8]]
=> ? = 13
[4,4,2,1,1,1]
=> [6,3,2,2]
=> [[1,2,7,11,12,13],[3,4,10],[5,6],[8,9]]
=> ? = 13
[4,3,3,1,1,1]
=> [6,3,3,1]
=> [[1,3,4,11,12,13],[2,6,7],[5,9,10],[8]]
=> ? = 12
[4,3,2,2,1,1]
=> [6,4,2,1]
=> [[1,3,6,7,12,13],[2,5,10,11],[4,9],[8]]
=> ? = 11
[4,2,2,2,2,1]
=> [6,5,1,1]
=> [[1,4,5,6,7,13],[2,9,10,11,12],[3],[8]]
=> ? = 10
[3,3,3,2,1,1]
=> [6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? = 10
[3,3,2,2,2,1]
=> [6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? = 9
[6,3,2,1,1,1]
=> [6,3,2,1,1,1]
=> [[1,5,8,12,13,14],[2,7,11],[3,10],[4],[6],[9]]
=> ? = 19
[6,2,2,2,1,1]
=> [6,4,1,1,1,1]
=> [[1,6,7,8,13,14],[2,10,11,12],[3],[4],[5],[9]]
=> ? = 18
[5,4,2,1,1,1]
=> [6,3,2,2,1]
=> [[1,3,8,12,13,14],[2,5,11],[4,7],[6,10],[9]]
=> ? = 17
[5,3,3,1,1,1]
=> [6,3,3,1,1]
=> [[1,4,5,12,13,14],[2,7,8],[3,10,11],[6],[9]]
=> ? = 16
[5,3,2,2,1,1]
=> [6,4,2,1,1]
=> [[1,4,7,8,13,14],[2,6,11,12],[3,10],[5],[9]]
=> ? = 15
[5,2,2,2,2,1]
=> [6,5,1,1,1]
=> [[1,5,6,7,8,14],[2,10,11,12,13],[3],[4],[9]]
=> ? = 14
[4,4,3,1,1,1]
=> [6,3,3,2]
=> [[1,2,5,12,13,14],[3,4,8],[6,7,11],[9,10]]
=> ? = 15
[4,4,2,2,1,1]
=> [6,4,2,2]
=> [[1,2,7,8,13,14],[3,4,11,12],[5,6],[9,10]]
=> ? = 14
[4,3,3,2,1,1]
=> [6,4,3,1]
=> [[1,3,4,8,13,14],[2,6,7,12],[5,10,11],[9]]
=> ? = 13
[4,3,2,2,2,1]
=> [6,5,2,1]
=> [[1,3,6,7,8,14],[2,5,11,12,13],[4,10],[9]]
=> ? = 12
[3,3,3,3,1,1]
=> [6,4,4]
=> [[1,2,3,4,13,14],[5,6,7,8],[9,10,11,12]]
=> ? = 12
[3,3,3,2,2,1]
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? = 11
[6,4,2,1,1,1]
=> [6,3,2,2,1,1]
=> [[1,4,9,13,14,15],[2,6,12],[3,8],[5,11],[7],[10]]
=> ? = 22
[6,3,3,1,1,1]
=> [6,3,3,1,1,1]
=> [[1,5,6,13,14,15],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 21
[6,3,2,2,1,1]
=> [6,4,2,1,1,1]
=> [[1,5,8,9,14,15],[2,7,12,13],[3,11],[4],[6],[10]]
=> ? = 20
[6,2,2,2,2,1]
=> [6,5,1,1,1,1]
=> [[1,6,7,8,9,15],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? = 19
[5,5,2,1,1,1]
=> [6,3,2,2,2]
=> [[1,2,9,13,14,15],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 21
[5,4,3,1,1,1]
=> [6,3,3,2,1]
=> [[1,3,6,13,14,15],[2,5,9],[4,8,12],[7,11],[10]]
=> ? = 19
[5,4,2,2,1,1]
=> [6,4,2,2,1]
=> [[1,3,8,9,14,15],[2,5,12,13],[4,7],[6,11],[10]]
=> ? = 18
[5,3,3,2,1,1]
=> [6,4,3,1,1]
=> [[1,4,5,9,14,15],[2,7,8,13],[3,11,12],[6],[10]]
=> ? = 17
[5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [[1,4,7,8,9,15],[2,6,12,13,14],[3,11],[5],[10]]
=> ? = 16
[4,4,4,1,1,1]
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 18
[4,4,3,2,1,1]
=> [6,4,3,2]
=> [[1,2,5,9,14,15],[3,4,8,13],[6,7,12],[10,11]]
=> ? = 16
[4,4,2,2,2,1]
=> [6,5,2,2]
=> [[1,2,7,8,9,15],[3,4,12,13,14],[5,6],[10,11]]
=> ? = 15
[4,3,3,3,1,1]
=> [6,4,4,1]
=> [[1,3,4,5,14,15],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 15
[4,3,3,2,2,1]
=> [6,5,3,1]
=> [[1,3,4,8,9,15],[2,6,7,13,14],[5,11,12],[10]]
=> ? = 14
[3,3,3,3,2,1]
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> ? = 13
[6,5,2,1,1,1]
=> [6,3,2,2,2,1]
=> [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
=> ? = 26
[6,4,3,1,1,1]
=> [6,3,3,2,1,1]
=> [[1,4,7,14,15,16],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? = 24
[6,4,2,2,1,1]
=> [6,4,2,2,1,1]
=> [[1,4,9,10,15,16],[2,6,13,14],[3,8],[5,12],[7],[11]]
=> ? = 23
[6,3,3,2,1,1]
=> [6,4,3,1,1,1]
=> [[1,5,6,10,15,16],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? = 22
[6,3,2,2,2,1]
=> [6,5,2,1,1,1]
=> [[1,5,8,9,10,16],[2,7,13,14,15],[3,12],[4],[6],[11]]
=> ? = 21
[5,5,3,1,1,1]
=> [6,3,3,2,2]
=> [[1,2,7,14,15,16],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? = 23
Description
The leg major index of a standard tableau. The leg length of a cell is the number of cells strictly below in the same column. This statistic is the sum of all leg lengths. Therefore, this is actually a statistic on the underlying integer partition. It happens to coincide with the (leg) major index of a tabloid restricted to standard Young tableaux, defined as follows: the descent set of a tabloid is the set of cells, not in the top row, whose entry is strictly larger than the entry directly above it. The leg major index is the sum of the leg lengths of the descents plus the number of descents.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000437: Permutations ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 10
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 6
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 10
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 7
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 11
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 10
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 9
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 6
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 6
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 13
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 11
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 10
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 12
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 9
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 8
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 7
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => 6
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 7
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 6
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => 4
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => ? = 4
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => ? = 7
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => ? = 6
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 5
[5,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,7,1] => ? = 10
[4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => ? = 9
[3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => ? = 9
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => ? = 8
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 7
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,6,1] => ? = 17
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => ? = 14
[5,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ? = 13
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,7,1] => ? = 13
[4,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => ? = 12
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => ? = 11
[4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => ? = 10
[3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => ? = 10
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 9
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,6,1] => ? = 19
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => ? = 18
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,7,1] => ? = 17
[5,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => ? = 16
[5,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ? = 15
[5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => ? = 14
[4,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => ? = 15
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => ? = 14
[4,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => ? = 13
[4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 12
[3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ? = 12
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => ? = 11
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,4,6,1] => ? = 22
[6,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,6,1] => ? = 21
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,6,1] => ? = 20
[6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 19
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => ? = 21
[5,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => ? = 19
[5,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => ? = 18
[5,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => ? = 17
[5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => ? = 16
[4,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ? = 18
[4,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => ? = 16
[4,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 15
[4,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => ? = 15
[4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => ? = 14
[3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 13
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,5,1] => ? = 26
[6,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,6,1] => ? = 24
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => ? = 23
[6,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => ? = 22
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00066: Permutations inversePermutations
St000436: Permutations ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 6
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 3
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 10
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,2,4,5,1] => 6
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 4
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [3,2,4,5,6,1] => 10
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => 7
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => 6
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => 6
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [6,2,1,3,4,5] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => 11
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [4,2,3,5,6,1] => 10
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 9
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 6
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => 6
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 4
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,2,3,1,4,5] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [6,3,1,2,4,5] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => 13
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [4,3,2,5,6,1] => 11
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [5,2,3,4,6,1] => 10
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => 12
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => 9
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => 8
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => 7
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [6,2,3,4,1,5] => 6
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => 7
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => 6
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => 5
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [6,3,2,1,4,5] => 4
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => ? = 4
[3,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,6,7,1] => [7,4,3,1,2,5,6] => ? = 7
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [7,5,2,1,3,4,6] => ? = 6
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 5
[5,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => [7,4,2,3,5,1,6] => ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,7,1] => [7,4,3,2,1,5,6] => ? = 10
[4,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => [7,5,2,3,1,4,6] => ? = 9
[3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [4,5,6,2,3,7,1] => [7,4,5,1,2,3,6] => ? = 9
[3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [7,5,3,1,2,4,6] => ? = 8
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => [7,6,2,1,3,4,5] => ? = 7
[6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,6,1] => [7,4,2,3,5,6,1] => ? = 17
[5,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => [7,4,3,2,5,1,6] => ? = 14
[5,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => [7,5,2,3,4,1,6] => ? = 13
[4,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,7,1] => [7,4,3,5,1,2,6] => ? = 13
[4,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [7,4,5,2,1,3,6] => ? = 12
[4,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [7,5,3,2,1,4,6] => ? = 11
[4,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,7,2,1] => [7,6,2,3,1,4,5] => ? = 10
[3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [7,5,4,1,2,3,6] => ? = 10
[3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [7,6,3,1,2,4,5] => ? = 9
[6,3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,6,1] => [7,4,3,2,5,6,1] => ? = 19
[6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => [7,5,2,3,4,6,1] => ? = 18
[5,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,7,1] => [7,4,3,5,2,1,6] => ? = 17
[5,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [7,4,5,2,3,1,6] => ? = 16
[5,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [7,5,3,2,4,1,6] => ? = 15
[5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,2,1] => [7,6,2,3,4,1,5] => ? = 14
[4,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [7,4,5,3,1,2,6] => ? = 15
[4,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [7,5,3,4,1,2,6] => ? = 14
[4,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => [7,5,4,2,1,3,6] => ? = 13
[4,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [7,6,3,2,1,4,5] => ? = 12
[3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [7,5,6,1,2,3,4] => ? = 12
[3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [7,6,4,1,2,3,5] => ? = 11
[6,4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,4,6,1] => [7,4,3,5,2,6,1] => ? = 22
[6,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,6,1] => [7,4,5,2,3,6,1] => ? = 21
[6,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,6,1] => [7,5,3,2,4,6,1] => ? = 20
[6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => [7,6,2,3,4,5,1] => ? = 19
[5,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [7,4,3,5,6,1,2] => ? = 21
[5,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => [7,4,5,3,2,1,6] => ? = 19
[5,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [7,5,3,4,2,1,6] => ? = 18
[5,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [7,5,4,2,3,1,6] => ? = 17
[5,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [7,6,3,2,4,1,5] => ? = 16
[4,4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [7,4,5,6,1,2,3] => ? = 18
[4,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [7,5,4,3,1,2,6] => ? = 16
[4,4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [7,6,3,4,1,2,5] => ? = 15
[4,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [7,5,6,2,1,3,4] => ? = 15
[4,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => [7,6,4,2,1,3,5] => ? = 14
[3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 13
[6,5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,5,1] => [7,4,3,5,6,2,1] => ? = 26
[6,4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,6,1] => [7,4,5,3,2,6,1] => ? = 24
[6,4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [7,5,3,4,2,6,1] => ? = 23
[6,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [7,5,4,2,3,6,1] => ? = 22
Description
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
Matching statistic: St000566
St000566: Integer partitions ⟶ ℤResult quality: 52% values known / values provided: 52%distinct values known / distinct values provided: 53%
Values
[1]
=> ? = 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 3
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 6
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 10
[4,1]
=> 6
[3,2]
=> 4
[3,1,1]
=> 3
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[5,1]
=> 10
[4,2]
=> 7
[4,1,1]
=> 6
[3,3]
=> 6
[3,2,1]
=> 4
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[5,2]
=> 11
[5,1,1]
=> 10
[4,3]
=> 9
[4,2,1]
=> 7
[4,1,1,1]
=> 6
[3,3,1]
=> 6
[3,2,2]
=> 5
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[5,3]
=> 13
[5,2,1]
=> 11
[5,1,1,1]
=> 10
[4,4]
=> 12
[4,3,1]
=> 9
[4,2,2]
=> 8
[4,2,1,1]
=> 7
[4,1,1,1,1]
=> 6
[3,3,2]
=> 7
[3,3,1,1]
=> 6
[3,2,2,1]
=> 5
[3,2,1,1,1]
=> 4
[2,2,2,2]
=> 4
[6,2,2,1,1,1]
=> ? = 17
[5,4,3,1]
=> ? = 19
[5,4,2,2]
=> ? = 18
[5,4,2,1,1]
=> ? = 17
[5,3,3,2]
=> ? = 17
[5,3,3,1,1]
=> ? = 16
[5,3,2,2,1]
=> ? = 15
[5,3,2,1,1,1]
=> ? = 14
[5,2,2,2,1,1]
=> ? = 13
[4,4,3,2]
=> ? = 16
[4,4,3,1,1]
=> ? = 15
[4,4,2,2,1]
=> ? = 14
[4,4,2,1,1,1]
=> ? = 13
[4,3,3,2,1]
=> ? = 13
[4,3,3,1,1,1]
=> ? = 12
[4,3,2,2,1,1]
=> ? = 11
[4,2,2,2,2,1]
=> ? = 10
[3,3,3,2,1,1]
=> ? = 10
[3,3,2,2,2,1]
=> ? = 9
[6,3,2,1,1,1]
=> ? = 19
[6,2,2,2,1,1]
=> ? = 18
[5,4,3,2]
=> ? = 20
[5,4,3,1,1]
=> ? = 19
[5,4,2,2,1]
=> ? = 18
[5,4,2,1,1,1]
=> ? = 17
[5,3,3,2,1]
=> ? = 17
[5,3,3,1,1,1]
=> ? = 16
[5,3,2,2,1,1]
=> ? = 15
[5,2,2,2,2,1]
=> ? = 14
[4,4,3,2,1]
=> ? = 16
[4,4,3,1,1,1]
=> ? = 15
[4,4,2,2,1,1]
=> ? = 14
[4,3,3,2,1,1]
=> ? = 13
[4,3,2,2,2,1]
=> ? = 12
[3,3,3,3,1,1]
=> ? = 12
[3,3,3,2,2,1]
=> ? = 11
[6,4,2,1,1,1]
=> ? = 22
[6,3,3,1,1,1]
=> ? = 21
[6,3,2,2,1,1]
=> ? = 20
[6,2,2,2,2,1]
=> ? = 19
[5,5,2,1,1,1]
=> ? = 21
[5,4,3,2,1]
=> ? = 20
[5,4,3,1,1,1]
=> ? = 19
[5,4,2,2,1,1]
=> ? = 18
[5,3,3,2,1,1]
=> ? = 17
[5,3,2,2,2,1]
=> ? = 16
[4,4,4,1,1,1]
=> ? = 18
[4,4,3,2,1,1]
=> ? = 16
[4,4,2,2,2,1]
=> ? = 15
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000391
Mp00044: Integer partitions conjugateInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00134: Standard tableaux descent wordBinary words
St000391: Binary words ⟶ ℤResult quality: 36% values known / values provided: 36%distinct values known / distinct values provided: 44%
Values
[1]
=> [1]
=> [[1]]
=> => ? = 0
[2]
=> [1,1]
=> [[1],[2]]
=> 1 => 1
[1,1]
=> [2]
=> [[1,2]]
=> 0 => 0
[3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 11 => 3
[2,1]
=> [2,1]
=> [[1,3],[2]]
=> 10 => 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 00 => 0
[4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 6
[3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 3
[2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 010 => 2
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 100 => 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 000 => 0
[5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 10
[4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 6
[3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 4
[3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 3
[2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0100 => 2
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0000 => 0
[5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 11110 => 10
[4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 11010 => 7
[4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 11100 => 6
[3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 01010 => 6
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 10100 => 4
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 11000 => 3
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 00100 => 3
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 01000 => 2
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 10000 => 1
[5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 111010 => 11
[5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 111100 => 10
[4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 101010 => 9
[4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 110100 => 7
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 111000 => 6
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 010100 => 6
[3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 100100 => 5
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 101000 => 4
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 110000 => 3
[2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 001000 => 3
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 010000 => 2
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 1101010 => 13
[5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> 1110100 => 11
[5,1,1,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> 1111000 => 10
[4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 0101010 => 12
[4,3,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 1010100 => 9
[4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 1100100 => 8
[4,2,1,1]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> 1101000 => 7
[4,1,1,1,1]
=> [5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> 1110000 => 6
[3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> 0100100 => 7
[3,3,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 0101000 => 6
[3,2,2,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 1001000 => 5
[3,2,1,1,1]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> 1010000 => 4
[2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 0001000 => 4
[5,4,2]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> 1010100100 => ? = 17
[5,4,1,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> 1010101000 => ? = 16
[5,3,3]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> 1100100100 => ? = 16
[5,3,2,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> 1101001000 => ? = 14
[5,3,1,1,1]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> 1101010000 => ? = 13
[5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> 1110001000 => ? = 13
[5,2,2,1,1]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> 1110010000 => ? = 12
[4,4,3]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> 0100100100 => ? = 15
[4,4,2,1]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> 0101001000 => ? = 13
[4,4,1,1,1]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> 0101010000 => ? = 12
[4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> 1001001000 => ? = 12
[4,3,2,2]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> 1010001000 => ? = 11
[4,3,2,1,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> 1010010000 => ? = 10
[4,2,2,2,1]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> 1100010000 => ? = 9
[3,3,2,2,1]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> 0100010000 => ? = 8
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 7
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 6
[2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 5
[5,4,3]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> 10100100100 => ? = 19
[5,4,2,1]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> 10101001000 => ? = 17
[5,4,1,1,1]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> 10101010000 => ? = 16
[5,3,3,1]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> 11001001000 => ? = 16
[5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> 11010001000 => ? = 15
[5,3,2,1,1]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> 11010010000 => ? = 14
[5,2,2,2,1]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> 11100010000 => ? = 13
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> [[1,5,6,10,11,12],[2,8,9],[3],[4],[7]]
=> ? => ? = 12
[4,4,3,1]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> 01001001000 => ? = 15
[4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 01010001000 => ? = 14
[4,4,2,1,1]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> 01010010000 => ? = 13
[4,3,3,2]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> 10010001000 => ? = 13
[4,3,3,1,1]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> 10010010000 => ? = 12
[4,3,2,2,1]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> 10100010000 => ? = 11
[4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? => ? = 10
[4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? => ? = 9
[3,3,3,2,1]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> 00100010000 => ? = 10
[3,3,3,1,1,1]
=> [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 9
[3,3,2,2,1,1]
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> 01000100000 => ? = 8
[3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? => ? = 7
[6,2,2,1,1,1]
=> [6,3,1,1,1,1]
=> [[1,6,7,11,12,13],[2,9,10],[3],[4],[5],[8]]
=> ? => ? = 17
[5,4,3,1]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> 101001001000 => ? = 19
[5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> 101010001000 => ? = 18
[5,4,2,1,1]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> 101010010000 => ? = 17
[5,3,3,2]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> 110010001000 => ? = 17
[5,3,3,1,1]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> 110010010000 => ? = 16
[5,3,2,2,1]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> 110100010000 => ? = 15
[5,3,2,1,1,1]
=> [6,3,2,1,1]
=> [[1,4,7,11,12,13],[2,6,10],[3,9],[5],[8]]
=> ? => ? = 14
[5,2,2,2,1,1]
=> [6,4,1,1,1]
=> [[1,5,6,7,12,13],[2,9,10,11],[3],[4],[8]]
=> ? => ? = 13
[4,4,3,2]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> 010010001000 => ? = 16
[4,4,3,1,1]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> 010010010000 => ? = 15
Description
The sum of the positions of the ones in a binary word.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000246The number of non-inversions of a permutation. St000009The charge of a standard tableau. St000008The major index of the composition. St001697The shifted natural comajor index of a standard Young tableau. St000018The number of inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000492The rob statistic of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000499The rcb statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000161The sum of the sizes of the right subtrees of a binary tree. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000004The major index of a permutation. St000305The inverse major index of a permutation. St000341The non-inversion sum of a permutation. St000446The disorder of a permutation. St001874Lusztig's a-function for the symmetric group. St000798The makl of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.