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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St000081
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Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2,1] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => ([],2)
=> 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> 1
[3,2,1] => [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 6
Description
The number of edges of a graph.
Matching statistic: St000391
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(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => => ? = 0
[1,2] => [2,1] => [2,1] => 1 => 1
[2,1] => [1,2] => [1,2] => 0 => 0
[1,2,3] => [3,2,1] => [3,2,1] => 11 => 3
[1,3,2] => [2,3,1] => [2,3,1] => 01 => 2
[2,1,3] => [3,1,2] => [1,3,2] => 01 => 2
[2,3,1] => [1,3,2] => [3,1,2] => 10 => 1
[3,1,2] => [2,1,3] => [2,1,3] => 10 => 1
[3,2,1] => [1,2,3] => [1,2,3] => 00 => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 111 => 6
[1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 011 => 5
[1,3,2,4] => [4,2,3,1] => [2,4,3,1] => 011 => 5
[1,3,4,2] => [2,4,3,1] => [4,2,3,1] => 101 => 4
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 101 => 4
[1,4,3,2] => [2,3,4,1] => [2,3,4,1] => 001 => 3
[2,1,3,4] => [4,3,1,2] => [1,4,3,2] => 011 => 5
[2,1,4,3] => [3,4,1,2] => [3,1,4,2] => 101 => 4
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 101 => 4
[2,3,4,1] => [1,4,3,2] => [4,3,1,2] => 110 => 3
[2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 001 => 3
[2,4,3,1] => [1,3,4,2] => [3,4,1,2] => 010 => 2
[3,1,2,4] => [4,2,1,3] => [2,1,4,3] => 101 => 4
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 110 => 3
[3,2,1,4] => [4,1,2,3] => [1,2,4,3] => 001 => 3
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 010 => 2
[3,4,1,2] => [2,1,4,3] => [2,4,1,3] => 010 => 2
[3,4,2,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 110 => 3
[4,1,3,2] => [2,3,1,4] => [2,3,1,4] => 010 => 2
[4,2,1,3] => [3,1,2,4] => [1,3,2,4] => 010 => 2
[4,2,3,1] => [1,3,2,4] => [3,1,2,4] => 100 => 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 100 => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 10
[1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => 0111 => 9
[1,2,4,3,5] => [5,3,4,2,1] => [3,5,4,2,1] => 0111 => 9
[1,2,4,5,3] => [3,5,4,2,1] => [5,3,4,2,1] => 1011 => 8
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,5,2,1] => 1011 => 8
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,5,2,1] => 0011 => 7
[1,3,2,4,5] => [5,4,2,3,1] => [2,5,4,3,1] => 0111 => 9
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,5,3,1] => 1011 => 8
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 1011 => 8
[1,3,4,5,2] => [2,5,4,3,1] => [5,4,2,3,1] => 1101 => 7
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,5,3,1] => 0011 => 7
[1,3,5,4,2] => [2,4,5,3,1] => [4,5,2,3,1] => 0101 => 6
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,5,4,1] => 1011 => 8
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,2,4,1] => 1101 => 7
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,5,4,1] => 0011 => 7
[1,4,3,5,2] => [2,5,3,4,1] => [2,5,3,4,1] => 0101 => 6
[1,4,5,2,3] => [3,2,5,4,1] => [3,5,2,4,1] => 0101 => 6
[1,4,5,3,2] => [2,3,5,4,1] => [5,2,3,4,1] => 1001 => 5
[1,2,4,5,3,7,8,6] => [6,8,7,3,5,4,2,1] => [6,8,3,7,5,4,2,1] => ? => ? = 24
[2,3,1,6,4,5,7,8] => [8,7,5,4,6,1,3,2] => [1,5,4,8,7,6,3,2] => ? => ? = 24
[1,3,4,2,7,5,6,8] => [8,6,5,7,2,4,3,1] => [2,6,5,8,7,4,3,1] => ? => ? = 24
[3,1,2,4,5,7,8,6] => [6,8,7,5,4,2,1,3] => [8,6,2,1,7,5,4,3] => ? => ? = 24
[1,2,6,3,8,4,5,7] => [7,5,4,8,3,6,2,1] => [5,4,7,3,8,6,2,1] => ? => ? = 22
[1,2,3,6,4,5,7,8] => [8,7,5,4,6,3,2,1] => [5,4,8,7,6,3,2,1] => ? => ? = 26
[] => [] => [] => ? => ? = 1
[1,2,3,4,5,7,8,9,6] => [6,9,8,7,5,4,3,2,1] => [9,8,6,7,5,4,3,2,1] => ? => ? = 33
[3,2,4,1,8,5,7,6] => [6,7,5,8,1,4,2,3] => [1,6,7,5,2,8,4,3] => ? => ? = 20
[4,3,1,2,8,6,7,5] => [5,7,6,8,2,1,3,4] => [7,5,2,1,6,3,8,4] => ? => ? = 18
[1,2,6,5,8,7,3,4] => [4,3,7,8,5,6,2,1] => [7,8,4,3,5,6,2,1] => ? => ? = 18
[1,5,2,3,4,6,7,8,9] => [9,8,7,6,4,3,2,5,1] => [4,3,2,9,8,7,6,5,1] => ? => ? = 33
[5,6,2,1,4,3,7,8] => [8,7,3,4,1,2,6,5] => [3,1,4,8,2,7,6,5] => ? => ? = 18
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000008
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 99%●distinct values known / distinct values provided: 88%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 88% ●values known / values provided: 99%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [1,2] => [2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1] => 3
[1,3,2] => [3,1,2] => [1,3,2] => [2,1] => 2
[2,1,3] => [2,3,1] => [2,3,1] => [2,1] => 2
[2,3,1] => [2,1,3] => [2,1,3] => [1,2] => 1
[3,1,2] => [1,3,2] => [3,1,2] => [1,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[1,2,4,3] => [4,3,1,2] => [1,4,3,2] => [2,1,1] => 5
[1,3,2,4] => [4,2,3,1] => [2,4,3,1] => [2,1,1] => 5
[1,3,4,2] => [4,2,1,3] => [2,1,4,3] => [1,2,1] => 4
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [1,2,1] => 4
[1,4,3,2] => [4,1,2,3] => [1,2,4,3] => [3,1] => 3
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => [2,1,1] => 5
[2,1,4,3] => [3,4,1,2] => [3,1,4,2] => [1,2,1] => 4
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => [1,2,1] => 4
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[2,4,1,3] => [3,1,4,2] => [1,3,4,2] => [3,1] => 3
[2,4,3,1] => [3,1,2,4] => [1,3,2,4] => [2,2] => 2
[3,1,2,4] => [2,4,3,1] => [4,2,3,1] => [1,2,1] => 4
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [1,1,2] => 3
[3,2,1,4] => [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[3,2,4,1] => [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[3,4,1,2] => [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [2,2] => 2
[4,2,1,3] => [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[4,2,3,1] => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[4,3,1,2] => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1] => 10
[1,2,3,5,4] => [5,4,3,1,2] => [1,5,4,3,2] => [2,1,1,1] => 9
[1,2,4,3,5] => [5,4,2,3,1] => [2,5,4,3,1] => [2,1,1,1] => 9
[1,2,4,5,3] => [5,4,2,1,3] => [2,1,5,4,3] => [1,2,1,1] => 8
[1,2,5,3,4] => [5,4,1,3,2] => [5,1,4,3,2] => [1,2,1,1] => 8
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,4,3] => [3,1,1] => 7
[1,3,2,4,5] => [5,3,4,2,1] => [3,5,4,2,1] => [2,1,1,1] => 9
[1,3,2,5,4] => [5,3,4,1,2] => [3,1,5,4,2] => [1,2,1,1] => 8
[1,3,4,2,5] => [5,3,2,4,1] => [3,2,5,4,1] => [1,2,1,1] => 8
[1,3,4,5,2] => [5,3,2,1,4] => [3,2,1,5,4] => [1,1,2,1] => 7
[1,3,5,2,4] => [5,3,1,4,2] => [1,3,5,4,2] => [3,1,1] => 7
[1,3,5,4,2] => [5,3,1,2,4] => [1,3,2,5,4] => [2,2,1] => 6
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,2,1,1] => 8
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,1,4,3] => [1,1,2,1] => 7
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,5,4,1] => [3,1,1] => 7
[1,4,3,5,2] => [5,2,3,1,4] => [2,3,1,5,4] => [2,2,1] => 6
[1,4,5,2,3] => [5,2,1,4,3] => [2,5,1,4,3] => [2,2,1] => 6
[] => [] => [] => [] => ? = 1
[1,2,3,4,5,6,7,9,8] => [9,8,7,6,5,4,3,1,2] => [1,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1] => ? = 35
[2,1,3,4,5,6,7,8,9] => [8,9,7,6,5,4,3,2,1] => [8,9,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1] => ? = 35
[1,2,3,4,5,6,7,8,10,9] => [10,9,8,7,6,5,4,3,1,2] => [1,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1] => ? = 44
[2,1,3,4,5,6,7,8,9,10] => [9,10,8,7,6,5,4,3,2,1] => [9,10,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
[1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1] => ? = 36
[1,2,3,4,5,7,8,9,6] => [9,8,7,6,5,3,2,1,4] => [3,2,1,9,8,7,6,5,4] => [1,1,2,1,1,1,1,1] => ? = 33
[1,2,3,4,5,9,6,7,8] => [9,8,7,6,5,1,4,3,2] => [9,8,1,7,6,5,4,3,2] => [1,1,2,1,1,1,1,1] => ? = 33
[1,2,4,3,5,6,7,8,9] => [9,8,6,7,5,4,3,2,1] => ? => ? => ? = 35
[1,2,3,4,5,6,8,7,9] => [9,8,7,6,5,4,2,3,1] => [2,9,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1,1] => ? = 35
[1,2,3,4,5,6,7,9,8,10] => [10,9,8,7,6,5,4,2,3,1] => [2,10,9,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
[1,3,2,4,5,6,7,8,9] => [9,7,8,6,5,4,3,2,1] => [7,9,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1] => ? = 35
[1,5,2,3,4,6,7,8,9] => [9,5,8,7,6,4,3,2,1] => ? => ? => ? = 33
[1,2,3,4,6,5,7,8,9] => [9,8,7,6,4,5,3,2,1] => [4,9,8,7,6,5,3,2,1] => [2,1,1,1,1,1,1,1] => ? = 35
[1,2,3,5,4,6,7,8,9] => [9,8,7,5,6,4,3,2,1] => [5,9,8,7,6,4,3,2,1] => [2,1,1,1,1,1,1,1] => ? = 35
[1,2,3,4,5,7,6,8,9,10] => [10,9,8,7,6,4,5,3,2,1] => ? => ? => ? = 44
[1,2,3,4,6,5,7,8,9,10] => [10,9,8,7,5,6,4,3,2,1] => [5,10,9,8,7,6,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
[1,2,3,7,4,5,6,8,9] => [9,8,7,3,6,5,4,2,1] => ? => ? => ? = 33
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000947
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 93%●distinct values known / distinct values provided: 85%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 93%●distinct values known / distinct values provided: 85%
Values
[1] => [1] => [.,.]
=> [1,0]
=> ? = 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 6
[1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 5
[1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 5
[1,3,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,4,2,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,4,3,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 5
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,3,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 4
[2,3,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,4,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,4,3,1] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 4
[3,1,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,1,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,1,3,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,3,1,2] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[1,2,4,3,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[1,2,4,5,3] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 8
[1,2,5,3,4] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 8
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,3,2,4,5] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[1,3,2,5,4] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 8
[1,3,4,2,5] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 8
[1,3,4,5,2] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 7
[1,3,5,2,4] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 7
[1,3,5,4,2] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 6
[1,4,2,3,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 8
[1,4,2,5,3] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,4,3,2,5] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,4,3,5,2] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 6
[1,4,5,2,3] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 6
[1,4,5,3,2] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[7,8,3,4,5,6,1,2] => [3,4,7,5,1,8,6,2] => [[.,[.,[[.,.],.]]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 8
[7,8,5,6,1,2,3,4] => [5,6,7,3,1,8,4,2] => [[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 8
[7,8,3,4,1,2,5,6] => [5,3,6,4,7,1,8,2] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 12
[7,8,1,2,3,4,5,6] => [3,4,5,6,7,1,8,2] => [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 16
[7,1,2,3,4,5,6,8] => [2,3,4,5,6,7,1,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 22
[5,6,3,4,1,2,7,8] => [5,3,1,6,4,2,7,8] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[4,1,2,3,5,6,7,8] => [2,3,4,1,5,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,3,4,5,6,7,8,2] => [2,3,4,5,6,8,1,7] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 22
[1,2,3,4,6,7,8,5] => [5,6,8,1,2,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,2,5,4,3,8,7,6] => [3,8,4,7,1,2,5,6] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 22
[1,2,4,5,3,7,8,6] => [3,4,6,8,1,2,5,7] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 24
[1,4,3,2,5,8,7,6] => [2,8,3,7,1,4,5,6] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 22
[1,2,3,5,6,7,4,8] => [4,5,7,1,2,3,6,8] => ?
=> ?
=> ? = 25
[1,4,3,2,7,6,5,8] => [2,7,3,6,1,4,5,8] => [[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 22
[1,3,4,2,6,7,5,8] => [2,3,5,7,1,4,6,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 24
[1,2,4,5,6,3,7,8] => [3,4,6,1,2,5,7,8] => ?
=> ?
=> ? = 25
[2,4,5,6,1,7,3,8] => [2,3,7,1,5,4,6,8] => ?
=> ?
=> ? = 20
[2,4,5,6,7,1,8,3] => [2,3,4,8,1,6,5,7] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 18
[2,5,1,7,3,4,8,6] => [8,5,1,2,6,3,4,7] => [[[.,.],.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 20
[2,6,7,1,3,4,8,5] => [8,4,5,1,2,6,3,7] => [[[.,.],[.,.]],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 18
[1,3,5,6,2,7,4,8] => [3,7,2,5,1,4,6,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 22
[1,3,5,6,7,2,8,4] => [3,4,8,2,6,1,5,7] => ?
=> ?
=> ? = 20
[3,6,7,1,2,8,4,5] => [7,8,1,4,2,5,3,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 16
[3,1,2,4,6,7,5,8] => [5,1,7,2,3,4,6,8] => ?
=> ?
=> ? = 24
[1,3,4,2,7,5,6,8] => ? => ?
=> ?
=> ? = 24
[1,3,4,2,5,8,6,7] => [2,3,7,8,1,4,5,6] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 24
[1,4,5,2,3,6,7,8] => [4,2,5,1,3,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[1,4,5,6,7,2,3,8] => [2,3,6,4,7,1,5,8] => [[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 20
[1,4,2,3,6,7,5,8] => [5,2,7,1,3,4,6,8] => ?
=> ?
=> ? = 24
[1,2,4,6,7,3,8,5] => [4,8,3,6,1,2,5,7] => ?
=> ?
=> ? = 22
[1,4,2,3,7,5,6,8] => [6,2,7,1,3,4,5,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[1,4,2,3,5,7,8,6] => [6,2,8,1,3,4,5,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[1,4,2,3,5,8,6,7] => [7,2,8,1,3,4,5,6] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[1,2,5,6,3,4,7,8] => [5,3,6,1,2,4,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[1,2,5,6,7,8,3,4] => [3,4,7,5,8,1,2,6] => [[.,[.,[[.,.],[.,.]]]],[.,[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 20
[1,5,2,7,3,4,6,8] => [5,6,2,7,1,3,4,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 22
[1,2,5,3,4,7,8,6] => [6,3,8,1,2,4,5,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[5,1,2,8,3,4,6,7] => [5,6,7,1,8,2,3,4] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 20
[1,2,5,3,4,8,6,7] => [7,3,8,1,2,4,5,6] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[1,5,2,3,4,6,7,8] => [3,4,5,1,2,6,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,6,7,2,3,4,5,8] => [4,5,6,2,7,1,3,8] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 20
[1,2,3,6,7,4,5,8] => [6,4,7,1,2,3,5,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 24
[6,1,8,2,3,4,5,7] => [4,5,6,7,1,8,2,3] => [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 18
[1,6,2,8,3,4,5,7] => [5,6,7,2,8,1,3,4] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 20
[1,2,6,3,8,4,5,7] => [6,7,3,8,1,2,4,5] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 22
[1,2,6,3,4,5,7,8] => [4,5,6,1,2,3,7,8] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,2,3,6,4,5,7,8] => [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 26
[1,2,7,8,3,4,5,6] => [5,6,7,3,8,1,2,4] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 20
Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St001397
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2,1] => ([],2)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> 0
[1,2,3] => [3,2,1] => ([],3)
=> 3
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> 2
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [4,3,2,1] => ([],4)
=> 6
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> 5
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 5
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 4
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 4
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> 5
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 4
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 4
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 4
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 3
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 10
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> 9
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 9
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 8
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 8
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 7
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 9
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 8
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 8
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 7
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> 7
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 6
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 8
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> 7
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 7
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 6
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 14
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 15
[1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 13
[1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 14
[1,2,6,5,7,4,3] => [3,4,7,5,6,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 13
[1,2,7,3,5,6,4] => [4,6,5,3,7,2,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 15
[1,2,7,3,6,5,4] => [4,5,6,3,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 14
[1,2,7,4,5,3,6] => [6,3,5,4,7,2,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 15
[1,2,7,5,4,3,6] => [6,3,4,5,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 14
[1,2,7,5,6,4,3] => [3,4,6,5,7,2,1] => ([(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 12
[1,2,7,6,3,5,4] => [4,5,3,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 13
[1,2,7,6,4,3,5] => [5,3,4,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 13
[1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 11
[1,3,5,6,2,7,4] => [4,7,2,6,5,3,1] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6)],7)
=> ? = 15
[1,3,5,6,7,2,4] => [4,2,7,6,5,3,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 14
[1,3,5,7,6,4,2] => [2,4,6,7,5,3,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 12
[1,3,6,5,4,2,7] => [7,2,4,5,6,3,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 14
[1,3,6,5,7,4,2] => [2,4,7,5,6,3,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 12
[1,3,6,7,4,2,5] => [5,2,4,7,6,3,1] => ([(1,5),(1,6),(2,3),(2,4),(4,5),(4,6)],7)
=> ? = 13
[1,3,6,7,5,4,2] => [2,4,5,7,6,3,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 11
[1,3,7,5,6,4,2] => [2,4,6,5,7,3,1] => ([(1,2),(1,5),(3,6),(4,6),(5,3),(5,4)],7)
=> ? = 11
[1,3,7,6,5,4,2] => [2,4,5,6,7,3,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 10
[1,4,5,6,2,3,7] => [7,3,2,6,5,4,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 15
[1,4,5,6,2,7,3] => [3,7,2,6,5,4,1] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 14
[1,4,5,7,6,3,2] => [2,3,6,7,5,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 11
[1,4,6,2,7,3,5] => [5,3,7,2,6,4,1] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 14
[1,4,6,5,3,2,7] => [7,2,3,5,6,4,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 13
[1,4,6,5,3,7,2] => [2,7,3,5,6,4,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 12
[1,4,6,5,7,3,2] => [2,3,7,5,6,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 11
[1,4,6,7,2,3,5] => [5,3,2,7,6,4,1] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 13
[1,4,6,7,5,3,2] => [2,3,5,7,6,4,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 10
[1,4,7,5,6,3,2] => [2,3,6,5,7,4,1] => ([(1,5),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ? = 10
[1,4,7,6,5,3,2] => [2,3,5,6,7,4,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
=> ? = 9
[1,5,2,4,3,6,7] => [7,6,3,4,2,5,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[1,5,3,2,4,6,7] => [7,6,4,2,3,5,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[1,5,4,3,6,2,7] => [7,2,6,3,4,5,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 14
[1,5,4,6,3,2,7] => [7,2,3,6,4,5,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 13
[1,5,4,6,3,7,2] => [2,7,3,6,4,5,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 12
[1,5,4,6,7,3,2] => [2,3,7,6,4,5,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 11
[1,5,6,2,4,7,3] => [3,7,4,2,6,5,1] => ([(1,5),(1,6),(2,3),(2,4),(4,5),(4,6)],7)
=> ? = 13
[1,5,6,2,7,3,4] => [4,3,7,2,6,5,1] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 13
[1,5,6,4,3,7,2] => [2,7,3,4,6,5,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 11
[1,5,6,4,7,3,2] => [2,3,7,4,6,5,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 10
[1,5,7,6,4,3,2] => [2,3,4,6,7,5,1] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? = 8
[1,6,2,4,5,3,7] => [7,3,5,4,2,6,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 15
[1,6,2,5,4,3,7] => [7,3,4,5,2,6,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 14
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St000798
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 61%●distinct values known / distinct values provided: 58%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 61%●distinct values known / distinct values provided: 58%
Values
[1] => [1] => [.,.]
=> [1] => ? = 0
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 3
[1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[3,1,2] => [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 6
[1,2,4,3] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 5
[1,3,2,4] => [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 5
[1,3,4,2] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[1,4,2,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[1,4,3,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 5
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[2,3,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[2,3,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3
[2,4,1,3] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3
[2,4,3,1] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[3,1,2,4] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 4
[3,1,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[3,2,4,1] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[3,4,1,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[3,4,2,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[4,1,2,3] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[4,1,3,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[4,2,1,3] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[4,2,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[4,3,1,2] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 10
[1,2,3,5,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 9
[1,2,4,3,5] => [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 9
[1,2,4,5,3] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 8
[1,2,5,3,4] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 8
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 7
[1,3,2,4,5] => [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 9
[1,3,2,5,4] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 8
[1,3,4,2,5] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 8
[1,3,4,5,2] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 7
[1,3,5,2,4] => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 7
[1,3,5,4,2] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 6
[1,4,2,3,5] => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 8
[1,4,2,5,3] => [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 7
[1,4,3,2,5] => [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 7
[1,4,3,5,2] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 6
[1,4,5,2,3] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 6
[1,4,5,3,2] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 5
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 21
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 20
[1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 20
[1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 20
[1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 18
[1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,3,6,7,5,4] => [4,7,6,1,2,3,5] => [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => ? = 16
[1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 18
[1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => ? = 16
[1,2,3,7,6,4,5] => [6,7,5,1,2,3,4] => [[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [2,1,3,7,6,5,4] => ? = 16
[1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 20
[1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 18
[1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 17
[1,2,4,6,7,5,3] => [4,7,3,6,1,2,5] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,2,5,6,4,3,7] => [3,6,5,1,2,4,7] => [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => ? = 16
[1,2,5,6,4,7,3] => [3,5,4,7,1,2,6] => [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [2,4,3,1,7,6,5] => ? = 15
[1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [2,4,3,1,7,6,5] => ? = 15
[1,2,5,6,7,4,3] => [3,4,7,6,1,2,5] => [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [3,4,2,1,7,6,5] => ? = 14
[1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 18
[1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => ? = 16
[1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,6,5,3,4,7] => [5,6,4,1,2,3,7] => [[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [2,1,3,7,6,5,4] => ? = 16
[1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,6,7,5,4,3] => [3,7,6,5,1,2,4] => [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [2,3,4,1,7,6,5] => ? = 12
[1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 17
[1,2,7,3,5,6,4] => [5,7,4,6,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,7,3,6,4,5] => [6,7,4,5,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,7,4,5,3,6] => [4,6,5,7,1,2,3] => [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [2,4,3,1,7,6,5] => ? = 15
[1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [3,4,2,1,7,6,5] => ? = 14
[1,2,7,5,3,4,6] => [5,6,4,7,1,2,3] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ? = 15
[1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [2,3,4,1,7,6,5] => ? = 12
[1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
=> [3,2,1,4,7,6,5] => ? = 14
[1,2,7,6,4,5,3] => [5,7,6,4,1,2,3] => [[[.,[[.,.],.]],.],[.,[.,.]]]
=> [2,3,1,4,7,6,5] => ? = 12
[1,2,7,6,5,3,4] => [6,7,5,4,1,2,3] => [[[[.,[.,.]],.],.],[.,[.,.]]]
=> [2,1,3,4,7,6,5] => ? = 12
[1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ? = 20
[1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 19
[1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 18
Description
The makl of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(1\underline{32})$, $(\underline{31}2)$, $(\underline{32}1)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => ([],1)
=> []
=> 0
[1,2] => [2,1] => ([(0,1)],2)
=> [1]
=> 1
[2,1] => [1,2] => ([],2)
=> []
=> 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> [2]
=> 2
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> [2]
=> 2
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> [1]
=> 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> [1]
=> 1
[3,2,1] => [1,2,3] => ([],3)
=> []
=> 0
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 6
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [2]
=> 2
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [2]
=> 2
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [1,1]
=> 2
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> [1]
=> 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [2]
=> 2
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [2]
=> 2
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> [1]
=> 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> [1]
=> 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> []
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 10
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> 9
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> 9
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 8
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 8
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> 9
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 8
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 8
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 8
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 6
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [15]
=> ? = 15
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [14]
=> ? = 14
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [14]
=> ? = 14
[1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> ? = 13
[1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> ? = 13
[1,2,3,6,5,4] => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,2,4,3,5,6] => [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)
=> [14]
=> ? = 14
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(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)
=> [13]
=> ? = 13
[1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> ? = 13
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,2,4,6,5,3] => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> ? = 13
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,2,5,4,3,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,2,5,4,6,3] => [3,6,4,5,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,2,6,3,5,4] => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,2,6,4,3,5] => [5,3,4,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,3,2,4,5,6] => [6,5,4,2,3,1] => ([(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)
=> [14]
=> ? = 14
[1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(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)
=> [13]
=> ? = 13
[1,3,2,5,4,6] => [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)
=> [13]
=> ? = 13
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,3,4,2,5,6] => [6,5,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> ? = 13
[1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,3,4,5,6,2] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,3,5,4,2,6] => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,4,2,3,5,6] => [6,5,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [13]
=> ? = 13
[1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,4,2,5,6,3] => [3,6,5,2,4,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,4,3,2,5,6] => [6,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,4,3,2,6,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,4,3,5,2,6] => [6,2,5,3,4,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,4,5,2,3,6] => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,5,2,3,4,6] => [6,4,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
[1,5,2,3,6,4] => [4,6,3,2,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,5,2,4,3,6] => [6,3,4,2,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,5,3,2,4,6] => [6,4,2,3,5,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
[2,1,3,4,5,6] => [6,5,4,3,1,2] => ([(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)
=> [14]
=> ? = 14
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000833
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 54%●distinct values known / distinct values provided: 48%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 54%●distinct values known / distinct values provided: 48%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,3,2] => [3,1,2] => [2,1,3] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 2
[2,3,1] => [1,3,2] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[1,2,4,3] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 5
[1,3,2,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 5
[1,3,4,2] => [2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 4
[1,4,2,3] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 4
[1,4,3,2] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 3
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 5
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [2,1,4,3] => 4
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 4
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 3
[2,4,1,3] => [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 3
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [1,3,2,4] => 2
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 4
[3,1,4,2] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 3
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 3
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 1
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[4,1,3,2] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 2
[4,2,3,1] => [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 1
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[1,2,3,5,4] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 9
[1,2,4,3,5] => [4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => 9
[1,2,4,5,3] => [3,5,1,2,4] => [4,2,1,5,3] => [3,2,1,5,4] => 8
[1,2,5,3,4] => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 8
[1,2,5,4,3] => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 7
[1,3,2,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => 9
[1,3,2,5,4] => [2,5,1,3,4] => [4,3,1,5,2] => [3,2,1,5,4] => 8
[1,3,4,2,5] => [2,4,1,3,5] => [5,3,1,4,2] => [5,2,1,4,3] => 8
[1,3,4,5,2] => [2,3,5,1,4] => [4,1,5,3,2] => [2,1,5,4,3] => 7
[1,3,5,2,4] => [2,4,5,1,3] => [3,1,5,4,2] => [2,1,5,4,3] => 7
[1,3,5,4,2] => [5,2,4,1,3] => [3,1,4,2,5] => [2,1,4,3,5] => 6
[1,4,2,3,5] => [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => 8
[1,4,2,5,3] => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3,5] => 7
[1,4,3,2,5] => [4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,3,4] => 7
[1,4,3,5,2] => [3,2,5,1,4] => [4,1,5,2,3] => [3,1,5,2,4] => 6
[1,4,5,2,3] => [4,2,5,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => 6
[1,4,5,3,2] => [2,5,4,1,3] => [3,1,4,5,2] => [2,1,3,5,4] => 5
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 21
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 20
[1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [7,5,4,3,2,1,6] => [7,5,4,3,2,1,6] => ? = 20
[1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [6,4,3,2,1,7,5] => [5,4,3,2,1,7,6] => ? = 19
[1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => ? = 19
[1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 18
[1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 20
[1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => [6,5,3,2,1,7,4] => [5,4,3,2,1,7,6] => ? = 19
[1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [7,5,3,2,1,6,4] => [7,4,3,2,1,6,5] => ? = 19
[1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [6,3,2,1,7,5,4] => [4,3,2,1,7,6,5] => ? = 18
[1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [5,3,2,1,6,4,7] => [4,3,2,1,6,5,7] => ? = 17
[1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => [7,4,3,2,1,6,5] => ? = 19
[1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [7,4,3,2,1,5,6] => [7,4,3,2,1,5,6] => ? = 18
[1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => [6,3,2,1,7,4,5] => [5,3,2,1,7,4,6] => ? = 17
[1,2,3,6,7,5,4] => [4,7,6,1,2,3,5] => [5,3,2,1,6,7,4] => [4,3,2,1,5,7,6] => ? = 16
[1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 18
[1,2,3,7,4,6,5] => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 17
[1,2,3,7,5,4,6] => [6,5,7,1,2,3,4] => [4,3,2,1,7,5,6] => [4,3,2,1,7,5,6] => ? = 17
[1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [4,3,2,1,6,7,5] => [4,3,2,1,5,7,6] => ? = 16
[1,2,3,7,6,4,5] => [6,7,5,1,2,3,4] => [4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 16
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 15
[1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [7,6,5,3,2,1,4] => [7,6,5,3,2,1,4] => ? = 20
[1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => [6,5,4,2,1,7,3] => [5,4,3,2,1,7,6] => ? = 19
[1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => [7,5,4,2,1,6,3] => [7,4,3,2,1,6,5] => ? = 19
[1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => [7,6,4,2,1,5,3] => [7,6,3,2,1,5,4] => ? = 19
[1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [7,5,2,1,6,4,3] => [7,3,2,1,6,5,4] => ? = 18
[1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [6,2,1,7,5,4,3] => [3,2,1,7,6,5,4] => ? = 17
[1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => [5,2,1,6,4,3,7] => [3,2,1,6,5,4,7] => ? = 16
[1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => [7,4,2,1,5,3,6] => [7,3,2,1,5,4,6] => ? = 17
[1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => [6,2,1,7,4,3,5] => [5,2,1,7,4,3,6] => ? = 16
[1,2,4,6,7,5,3] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => [3,2,1,5,4,7,6] => ? = 15
[1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [4,2,1,6,3,7,5] => [3,2,1,5,4,7,6] => ? = 15
[1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => [4,2,1,5,3,6,7] => [3,2,1,5,4,6,7] => ? = 14
[1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => [7,6,3,2,1,5,4] => [7,6,3,2,1,5,4] => ? = 19
[1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => [7,6,3,2,1,4,5] => [7,6,3,2,1,4,5] => ? = 18
[1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => [7,5,2,1,6,3,4] => [7,4,2,1,6,3,5] => ? = 17
[1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => [6,2,1,7,5,3,4] => [4,2,1,7,6,3,5] => ? = 16
[1,2,5,6,4,3,7] => [3,6,5,1,2,4,7] => [7,4,2,1,5,6,3] => [7,3,2,1,4,6,5] => ? = 16
[1,2,5,6,4,7,3] => [3,5,4,7,1,2,6] => [6,2,1,7,4,5,3] => [4,2,1,7,3,6,5] => ? = 15
[1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [5,2,1,7,4,6,3] => [4,2,1,7,3,6,5] => ? = 15
[1,2,5,6,7,4,3] => [3,4,7,6,1,2,5] => [5,2,1,6,7,4,3] => [3,2,1,4,7,6,5] => ? = 14
[1,2,5,7,6,4,3] => [7,3,6,5,1,2,4] => [4,2,1,5,6,3,7] => [3,2,1,4,6,5,7] => ? = 13
[1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => [7,3,2,1,6,5,4] => ? = 18
[1,2,6,3,5,4,7] => [6,4,5,1,2,3,7] => [7,3,2,1,5,4,6] => [7,3,2,1,5,4,6] => ? = 17
[1,2,6,4,3,5,7] => [5,4,6,1,2,3,7] => [7,3,2,1,6,4,5] => [7,3,2,1,6,4,5] => ? = 17
[1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [7,3,2,1,5,6,4] => [7,3,2,1,4,6,5] => ? = 16
[1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [6,2,1,7,3,5,4] => [4,2,1,7,3,6,5] => ? = 15
[1,2,6,5,3,4,7] => [5,6,4,1,2,3,7] => [7,3,2,1,4,6,5] => [7,3,2,1,4,6,5] => ? = 16
[1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => [7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => ? = 15
Description
The comajor index of a permutation.
This is, $\operatorname{comaj}(\pi) = \sum_{i \in \operatorname{Des}(\pi)} (n-i)$ for a permutation $\pi$ of length $n$.
Matching statistic: St000446
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 50%●distinct values known / distinct values provided: 48%
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 50%●distinct values known / distinct values provided: 48%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 3
[1,3,2] => [3,1,2] => [2,1,3] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,1,2] => [2,3,1] => 2
[2,3,1] => [1,3,2] => [2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[1,2,4,3] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 5
[1,3,2,4] => [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 5
[1,3,4,2] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 4
[1,4,2,3] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 4
[1,4,3,2] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 3
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 5
[2,1,4,3] => [1,4,2,3] => [3,2,4,1] => [4,2,1,3] => 4
[2,3,1,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 4
[2,3,4,1] => [1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 3
[2,4,1,3] => [1,3,4,2] => [2,4,3,1] => [4,1,3,2] => 3
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => [3,1,2,4] => 2
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 4
[3,1,4,2] => [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 3
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 3
[3,2,4,1] => [2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 2
[3,4,1,2] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,4,2,1] => [1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 1
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[4,1,3,2] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,3,4,2] => 2
[4,2,3,1] => [2,4,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 10
[1,2,3,5,4] => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 9
[1,2,4,3,5] => [4,1,2,3,5] => [5,3,2,1,4] => [4,3,2,5,1] => 9
[1,2,4,5,3] => [3,5,1,2,4] => [4,2,1,5,3] => [3,2,5,1,4] => 8
[1,2,5,3,4] => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 8
[1,2,5,4,3] => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 7
[1,3,2,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 9
[1,3,2,5,4] => [2,5,1,3,4] => [4,3,1,5,2] => [3,5,2,1,4] => 8
[1,3,4,2,5] => [2,4,1,3,5] => [5,3,1,4,2] => [3,5,2,4,1] => 8
[1,3,4,5,2] => [2,3,5,1,4] => [4,1,5,3,2] => [2,5,4,1,3] => 7
[1,3,5,2,4] => [2,4,5,1,3] => [3,1,5,4,2] => [2,5,1,4,3] => 7
[1,3,5,4,2] => [5,2,4,1,3] => [3,1,4,2,5] => [2,4,1,3,5] => 6
[1,4,2,3,5] => [3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => 8
[1,4,2,5,3] => [5,3,1,2,4] => [4,2,1,3,5] => [3,2,4,1,5] => 7
[1,4,3,2,5] => [4,3,1,2,5] => [5,2,1,3,4] => [3,2,4,5,1] => 7
[1,4,3,5,2] => [3,2,5,1,4] => [4,1,5,2,3] => [2,4,5,1,3] => 6
[1,4,5,2,3] => [4,2,5,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => 6
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 21
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 20
[1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [7,5,4,3,2,1,6] => [6,5,4,3,2,7,1] => ? = 20
[1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [6,4,3,2,1,7,5] => [5,4,3,2,7,1,6] => ? = 19
[1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => ? = 19
[1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 18
[1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [7,6,4,3,2,1,5] => [6,5,4,3,7,2,1] => ? = 20
[1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => [6,5,3,2,1,7,4] => [5,4,3,7,2,1,6] => ? = 19
[1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [7,5,3,2,1,6,4] => [5,4,3,7,2,6,1] => ? = 19
[1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [6,3,2,1,7,5,4] => [4,3,2,7,6,1,5] => ? = 18
[1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [5,3,2,1,6,4,7] => [4,3,2,6,1,5,7] => ? = 17
[1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => [5,4,3,2,7,6,1] => ? = 19
[1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [7,4,3,2,1,5,6] => [5,4,3,2,6,7,1] => ? = 18
[1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => [6,3,2,1,7,4,5] => [4,3,2,6,7,1,5] => ? = 17
[1,2,3,6,7,5,4] => [4,7,6,1,2,3,5] => [5,3,2,1,6,7,4] => [4,3,2,7,1,5,6] => ? = 16
[1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 18
[1,2,3,7,4,6,5] => [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 17
[1,2,3,7,5,4,6] => [6,5,7,1,2,3,4] => [4,3,2,1,7,5,6] => [4,3,2,1,6,7,5] => ? = 17
[1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [4,3,2,1,6,7,5] => [4,3,2,1,7,5,6] => ? = 16
[1,2,3,7,6,4,5] => [6,7,5,1,2,3,4] => [4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 16
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 15
[1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [7,6,5,3,2,1,4] => [6,5,4,7,3,2,1] => ? = 20
[1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => [6,5,4,2,1,7,3] => [5,4,7,3,2,1,6] => ? = 19
[1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => [7,5,4,2,1,6,3] => [5,4,7,3,2,6,1] => ? = 19
[1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => [7,6,4,2,1,5,3] => [5,4,7,3,6,2,1] => ? = 19
[1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [7,5,2,1,6,4,3] => [4,3,7,6,2,5,1] => ? = 18
[1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [6,2,1,7,5,4,3] => [3,2,7,6,5,1,4] => ? = 17
[1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => [5,2,1,6,4,3,7] => [3,2,6,5,1,4,7] => ? = 16
[1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => [7,4,2,1,5,3,6] => [4,3,6,2,5,7,1] => ? = 17
[1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => [6,2,1,7,4,3,5] => [3,2,6,5,7,1,4] => ? = 16
[1,2,4,6,7,5,3] => [4,7,3,6,1,2,5] => [5,2,1,6,3,7,4] => [3,2,5,7,1,4,6] => ? = 15
[1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [4,2,1,6,3,7,5] => [3,2,5,1,7,4,6] => ? = 15
[1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => [4,2,1,5,3,6,7] => [3,2,5,1,4,6,7] => ? = 14
[1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => [7,6,3,2,1,5,4] => [5,4,3,7,6,2,1] => ? = 19
[1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => [7,6,3,2,1,4,5] => [5,4,3,6,7,2,1] => ? = 18
[1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => [7,5,2,1,6,3,4] => [4,3,6,7,2,5,1] => ? = 17
[1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => [6,2,1,7,5,3,4] => [3,2,6,7,5,1,4] => ? = 16
[1,2,5,6,4,3,7] => [3,6,5,1,2,4,7] => [7,4,2,1,5,6,3] => [4,3,7,2,5,6,1] => ? = 16
[1,2,5,6,4,7,3] => [3,5,4,7,1,2,6] => [6,2,1,7,4,5,3] => [3,2,7,5,6,1,4] => ? = 15
[1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [5,2,1,7,4,6,3] => [3,2,7,5,1,6,4] => ? = 15
[1,2,5,6,7,4,3] => [3,4,7,6,1,2,5] => [5,2,1,6,7,4,3] => [3,2,7,6,1,4,5] => ? = 14
[1,2,5,7,6,4,3] => [7,3,6,5,1,2,4] => [4,2,1,5,6,3,7] => [3,2,6,1,4,5,7] => ? = 13
[1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => [4,3,2,7,6,5,1] => ? = 18
[1,2,6,3,5,4,7] => [6,4,5,1,2,3,7] => [7,3,2,1,5,4,6] => [4,3,2,6,5,7,1] => ? = 17
[1,2,6,4,3,5,7] => [5,4,6,1,2,3,7] => [7,3,2,1,6,4,5] => [4,3,2,6,7,5,1] => ? = 17
[1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [7,3,2,1,5,6,4] => [4,3,2,7,5,6,1] => ? = 16
[1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [6,2,1,7,3,5,4] => [3,2,5,7,6,1,4] => ? = 15
[1,2,6,5,3,4,7] => [5,6,4,1,2,3,7] => [7,3,2,1,4,6,5] => [4,3,2,5,7,6,1] => ? = 16
[1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => ? = 15
[1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => [6,2,1,7,3,4,5] => [3,2,5,6,7,1,4] => ? = 14
Description
The disorder of a permutation.
Consider a permutation $\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning $\pi$ from left to right and remove the elements $1$ through $n$ on this order one after the other. The '''disorder''' of $\pi$ is defined to be the number of times a position was not removed in this process.
For example, the disorder of $[3,5,2,1,4]$ is $8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.
Matching statistic: St000795
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000795: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 49%●distinct values known / distinct values provided: 48%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000795: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 49%●distinct values known / distinct values provided: 48%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [2,3,1] => 3
[1,3,2] => [3,1,2] => [3,1,2] => 2
[2,1,3] => [2,3,1] => [3,2,1] => 2
[2,3,1] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [3,2,4,1] => 6
[1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 5
[1,3,2,4] => [4,2,3,1] => [2,3,4,1] => 5
[1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 4
[1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 4
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 3
[2,1,3,4] => [3,4,2,1] => [4,2,3,1] => 5
[2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 4
[2,3,1,4] => [3,2,4,1] => [2,4,3,1] => 4
[2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 3
[2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 3
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,2,4] => [2,4,3,1] => [3,4,2,1] => 4
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 3
[3,2,1,4] => [2,3,4,1] => [4,3,2,1] => 3
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 3
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [3,4,2,5,1] => 10
[1,2,3,5,4] => [5,4,3,1,2] => [3,4,1,5,2] => 9
[1,2,4,3,5] => [5,4,2,3,1] => [4,2,3,5,1] => 9
[1,2,4,5,3] => [5,4,2,1,3] => [4,2,5,1,3] => 8
[1,2,5,3,4] => [5,4,1,3,2] => [4,1,3,5,2] => 8
[1,2,5,4,3] => [5,4,1,2,3] => [4,1,5,2,3] => 7
[1,3,2,4,5] => [5,3,4,2,1] => [4,3,2,5,1] => 9
[1,3,2,5,4] => [5,3,4,1,2] => [4,3,1,5,2] => 8
[1,3,4,2,5] => [5,3,2,4,1] => [3,2,4,5,1] => 8
[1,3,4,5,2] => [5,3,2,1,4] => [3,2,5,1,4] => 7
[1,3,5,2,4] => [5,3,1,4,2] => [3,1,4,5,2] => 7
[1,3,5,4,2] => [5,3,1,2,4] => [3,1,5,2,4] => 6
[1,4,2,3,5] => [5,2,4,3,1] => [2,4,3,5,1] => 8
[1,4,2,5,3] => [5,2,4,1,3] => [2,4,1,5,3] => 7
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,5,1] => 7
[1,4,3,5,2] => [5,2,3,1,4] => [2,3,5,1,4] => 6
[1,4,5,2,3] => [5,2,1,4,3] => [2,4,5,1,3] => 6
[1,4,5,3,2] => [5,2,1,3,4] => [2,5,1,3,4] => 5
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 21
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [4,5,3,6,1,7,2] => ? = 20
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [4,5,2,6,3,7,1] => ? = 20
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [4,5,2,6,1,7,3] => ? = 19
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [4,5,1,6,3,7,2] => ? = 19
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [4,5,1,6,2,7,3] => ? = 18
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [5,3,4,6,2,7,1] => ? = 20
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [5,3,4,6,1,7,2] => ? = 19
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [5,3,6,2,4,7,1] => ? = 19
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [5,3,6,2,7,1,4] => ? = 18
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [5,3,6,1,7,2,4] => ? = 17
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [5,2,4,6,3,7,1] => ? = 19
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => [5,2,6,3,4,7,1] => ? = 18
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => [5,2,6,3,7,1,4] => ? = 17
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => [5,2,6,1,7,3,4] => ? = 16
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [5,1,4,6,3,7,2] => ? = 18
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [5,1,4,6,2,7,3] => ? = 17
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [5,1,6,3,4,7,2] => ? = 17
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [5,1,6,3,7,2,4] => ? = 16
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [5,1,6,2,4,7,3] => ? = 16
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [5,1,6,2,7,3,4] => ? = 15
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [5,4,3,6,2,7,1] => ? = 20
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [5,4,3,6,1,7,2] => ? = 19
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [5,4,2,6,3,7,1] => ? = 19
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [4,3,5,6,2,7,1] => ? = 19
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [4,3,6,2,5,7,1] => ? = 18
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [4,3,6,2,7,1,5] => ? = 17
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [4,3,6,1,7,2,5] => ? = 16
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => [4,2,6,3,5,7,1] => ? = 17
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => [4,2,6,3,7,1,5] => ? = 16
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => [4,2,6,1,7,3,5] => ? = 15
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [4,1,6,3,7,2,5] => ? = 15
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [4,1,6,2,7,3,5] => ? = 14
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [3,5,4,6,2,7,1] => ? = 19
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [3,4,5,6,2,7,1] => ? = 18
[1,2,5,4,6,3,7] => [7,6,3,4,2,5,1] => [3,4,6,2,5,7,1] => ? = 17
[1,2,5,4,6,7,3] => [7,6,3,4,2,1,5] => [3,4,6,2,7,1,5] => ? = 16
[1,2,5,6,4,3,7] => [7,6,3,2,4,5,1] => [3,6,2,4,5,7,1] => ? = 16
[1,2,5,6,4,7,3] => [7,6,3,2,4,1,5] => [3,6,2,4,7,1,5] => ? = 15
[1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => [3,6,2,5,7,1,4] => ? = 15
[1,2,5,6,7,4,3] => [7,6,3,2,1,4,5] => [3,6,2,7,1,4,5] => ? = 14
[1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => [3,6,1,7,2,4,5] => ? = 13
[1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => [5,4,6,2,3,7,1] => ? = 18
[1,2,6,3,5,4,7] => [7,6,2,5,3,4,1] => [5,2,3,6,4,7,1] => ? = 17
[1,2,6,4,3,5,7] => [7,6,2,4,5,3,1] => [4,5,6,2,3,7,1] => ? = 17
[1,2,6,4,5,3,7] => [7,6,2,4,3,5,1] => [4,6,2,3,5,7,1] => ? = 16
[1,2,6,4,5,7,3] => [7,6,2,4,3,1,5] => [4,6,2,3,7,1,5] => ? = 15
[1,2,6,5,3,4,7] => [7,6,2,3,5,4,1] => [5,6,2,3,4,7,1] => ? = 16
[1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => [6,2,3,4,5,7,1] => ? = 15
Description
The mad of a permutation.
According to [1], this is the sum of twice the number of occurrences of the vincular pattern of $(2\underline{31})$ plus the number of occurrences of the vincular patterns $(\underline{31}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000246The number of non-inversions of a permutation. St000004The major index of a permutation. St000154The sum of the descent bottoms of a permutation. St000304The load of a permutation. St000305The inverse major index of a permutation. St000796The stat' of a permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St001428The number of B-inversions of a signed permutation. St001622The number of join-irreducible elements of a lattice. St001621The number of atoms of a lattice. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001866The nesting alignments of a signed permutation. St001862The number of crossings of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001822The number of alignments of a signed permutation. St000441The number of successions of a permutation. St001433The flag major index of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001877Number of indecomposable injective modules with projective dimension 2.
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