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Your data matches 31 different statistics following compositions of up to 3 maps.
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Mp00066: Permutations inversePermutations
Mp00067: Permutations Foata bijectionPermutations
St000018: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [2,3,1] => 2
[3,2,1] => [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 3
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 4
[2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 1
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 4
[2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 4
[3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,4,1,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 3
[3,4,1,2] => [3,4,1,2] => [1,3,4,2] => 2
[3,4,2,1] => [4,3,1,2] => [1,4,3,2] => 3
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => 5
[4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 4
[4,2,3,1] => [4,2,3,1] => [2,4,3,1] => 4
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 5
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[1,2,4,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => 3
[1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,5,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 6
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,2,5,3,4] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [4,5,1,2,3] => 6
[1,3,5,4,2] => [1,5,2,4,3] => [5,1,4,2,3] => 6
[1,4,2,3,5] => [1,3,4,2,5] => [3,1,4,2,5] => 3
[1,4,2,5,3] => [1,3,5,2,4] => [3,1,2,5,4] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 5
[1,4,3,5,2] => [1,5,3,2,4] => [3,5,1,2,4] => 5
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 3
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions (i,i+1) needed to write π. Thus, it is also the Coxeter length of π.
Matching statistic: St000391
Mp00066: Permutations inversePermutations
Mp00109: Permutations descent wordBinary words
St000391: Binary words ⟶ ℤResult quality: 98% values known / values provided: 99%distinct values known / distinct values provided: 98%
Values
[1] => [1] => => ? = 0
[1,2] => [1,2] => 0 => 0
[2,1] => [2,1] => 1 => 1
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 2
[2,1,3] => [2,1,3] => 10 => 1
[2,3,1] => [3,1,2] => 10 => 1
[3,1,2] => [2,3,1] => 01 => 2
[3,2,1] => [3,2,1] => 11 => 3
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 3
[1,3,2,4] => [1,3,2,4] => 010 => 2
[1,3,4,2] => [1,4,2,3] => 010 => 2
[1,4,2,3] => [1,3,4,2] => 001 => 3
[1,4,3,2] => [1,4,3,2] => 011 => 5
[2,1,3,4] => [2,1,3,4] => 100 => 1
[2,1,4,3] => [2,1,4,3] => 101 => 4
[2,3,1,4] => [3,1,2,4] => 100 => 1
[2,3,4,1] => [4,1,2,3] => 100 => 1
[2,4,1,3] => [3,1,4,2] => 101 => 4
[2,4,3,1] => [4,1,3,2] => 101 => 4
[3,1,2,4] => [2,3,1,4] => 010 => 2
[3,1,4,2] => [2,4,1,3] => 010 => 2
[3,2,1,4] => [3,2,1,4] => 110 => 3
[3,2,4,1] => [4,2,1,3] => 110 => 3
[3,4,1,2] => [3,4,1,2] => 010 => 2
[3,4,2,1] => [4,3,1,2] => 110 => 3
[4,1,2,3] => [2,3,4,1] => 001 => 3
[4,1,3,2] => [2,4,3,1] => 011 => 5
[4,2,1,3] => [3,2,4,1] => 101 => 4
[4,2,3,1] => [4,2,3,1] => 101 => 4
[4,3,1,2] => [3,4,2,1] => 011 => 5
[4,3,2,1] => [4,3,2,1] => 111 => 6
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 4
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 3
[1,2,4,5,3] => [1,2,5,3,4] => 0010 => 3
[1,2,5,3,4] => [1,2,4,5,3] => 0001 => 4
[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 7
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 2
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 6
[1,3,4,2,5] => [1,4,2,3,5] => 0100 => 2
[1,3,4,5,2] => [1,5,2,3,4] => 0100 => 2
[1,3,5,2,4] => [1,4,2,5,3] => 0101 => 6
[1,3,5,4,2] => [1,5,2,4,3] => 0101 => 6
[1,4,2,3,5] => [1,3,4,2,5] => 0010 => 3
[1,4,2,5,3] => [1,3,5,2,4] => 0010 => 3
[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 5
[1,4,3,5,2] => [1,5,3,2,4] => 0110 => 5
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 3
[1,4,5,3,2] => [1,5,4,2,3] => 0110 => 5
[] => [] => ? => ? = 0
[12,11,10,9,8,7,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 11111111111 => ? = 66
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => 1000000000 => ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => 1111111110 => ? = 45
[2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => 1000000000 => ? = 1
[2,4,6,8,10,12,1,3,5,7,9,11] => [7,1,8,2,9,3,10,4,11,5,12,6] => 10101010101 => ? = 36
[10,11,9,8,7,6,5,4,3,2,1] => [11,10,9,8,7,6,5,4,3,1,2] => 1111111110 => ? = 45
[11,9,7,5,3,1,12,10,8,6,4,2] => [6,12,5,11,4,10,3,9,2,8,1,7] => 01010101010 => ? = 30
[7,1,8,2,9,3,10,4,11,5,12,6] => [2,4,6,8,10,12,1,3,5,7,9,11] => 00000100000 => ? = 6
[7,6,8,5,9,4,10,3,11,2,12,1] => [12,10,8,6,4,2,1,3,5,7,9,11] => 11111100000 => ? = 21
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000008
Mp00066: Permutations inversePermutations
Mp00071: Permutations descent compositionInteger compositions
St000008: Integer compositions ⟶ ℤResult quality: 79% values known / values provided: 96%distinct values known / distinct values provided: 79%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2] => 0
[2,1] => [2,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [3] => 0
[1,3,2] => [1,3,2] => [2,1] => 2
[2,1,3] => [2,1,3] => [1,2] => 1
[2,3,1] => [3,1,2] => [1,2] => 1
[3,1,2] => [2,3,1] => [2,1] => 2
[3,2,1] => [3,2,1] => [1,1,1] => 3
[1,2,3,4] => [1,2,3,4] => [4] => 0
[1,2,4,3] => [1,2,4,3] => [3,1] => 3
[1,3,2,4] => [1,3,2,4] => [2,2] => 2
[1,3,4,2] => [1,4,2,3] => [2,2] => 2
[1,4,2,3] => [1,3,4,2] => [3,1] => 3
[1,4,3,2] => [1,4,3,2] => [2,1,1] => 5
[2,1,3,4] => [2,1,3,4] => [1,3] => 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => 4
[2,3,1,4] => [3,1,2,4] => [1,3] => 1
[2,3,4,1] => [4,1,2,3] => [1,3] => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 4
[2,4,3,1] => [4,1,3,2] => [1,2,1] => 4
[3,1,2,4] => [2,3,1,4] => [2,2] => 2
[3,1,4,2] => [2,4,1,3] => [2,2] => 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[3,2,4,1] => [4,2,1,3] => [1,1,2] => 3
[3,4,1,2] => [3,4,1,2] => [2,2] => 2
[3,4,2,1] => [4,3,1,2] => [1,1,2] => 3
[4,1,2,3] => [2,3,4,1] => [3,1] => 3
[4,1,3,2] => [2,4,3,1] => [2,1,1] => 5
[4,2,1,3] => [3,2,4,1] => [1,2,1] => 4
[4,2,3,1] => [4,2,3,1] => [1,2,1] => 4
[4,3,1,2] => [3,4,2,1] => [2,1,1] => 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => 3
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => 3
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => 6
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => 6
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => 6
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => 3
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => 5
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => 5
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => 3
[9,3,1,2,4,5,6,7,8] => [3,4,2,5,6,7,8,9,1] => [2,6,1] => ? = 10
[] => [] => [] => ? = 0
[1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9] => ? = 0
[1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,9,7,8] => [7,2] => ? = 7
[1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ? = 0
[1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,10,8,9] => [8,2] => ? = 8
[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
[8,9,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,2] => ? = 28
[8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ? = 28
[9,10,8,7,6,5,4,3,2,1] => [10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,2] => ? = 36
[8,7,6,5,4,3,1,2,9] => [7,8,6,5,4,3,2,1,9] => [2,1,1,1,1,1,2] => ? = 27
[9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => [1,1,1,1,1,1,1,1,2] => ? = 36
[1,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1] => ? = 35
[1,10,9,8,7,6,5,4,3,2] => [1,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1] => ? = 44
[10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,2] => ? = 45
[2,4,6,8,10,1,3,5,7,9] => [6,1,7,2,8,3,9,4,10,5] => [1,2,2,2,2,1] => ? = 25
[2,4,6,8,10,12,1,3,5,7,9,11] => [7,1,8,2,9,3,10,4,11,5,12,6] => [1,2,2,2,2,2,1] => ? = 36
[1,2,3,4,5,8,6,9,7] => [1,2,3,4,5,7,9,6,8] => [7,2] => ? = 7
[8,1,2,3,4,5,6,7,9] => [2,3,4,5,6,7,8,1,9] => [7,2] => ? = 7
[9,1,2,3,4,5,6,7,8,10] => [2,3,4,5,6,7,8,9,1,10] => [8,2] => ? = 8
[7,1,2,3,4,5,6,8,9] => [2,3,4,5,6,7,1,8,9] => [6,3] => ? = 6
[3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => [1,1,7] => ? = 3
[4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => [1,1,1,6] => ? = 6
[5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,5] => ? = 10
[6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,4] => ? = 15
[7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => [1,1,1,1,1,1,3] => ? = 21
[3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => [1,1,8] => ? = 3
[4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => [1,1,1,7] => ? = 6
[5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => [1,1,1,1,6] => ? = 10
[6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => [1,1,1,1,1,5] => ? = 15
[7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => [1,1,1,1,1,1,4] => ? = 21
[8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9,10] => [1,1,1,1,1,1,1,3] => ? = 28
[10,11,9,8,7,6,5,4,3,2,1] => [11,10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,1,2] => ? = 45
[1,2,3,4,5,10,9,8,7,6] => [1,2,3,4,5,10,9,8,7,6] => [6,1,1,1,1] => ? = 30
[2,4,1,3,5,6,7,8,9] => [3,1,4,2,5,6,7,8,9] => [1,2,6] => ? = 4
[4,8,3,7,2,6,1,5,9] => [7,5,3,1,8,6,4,2,9] => [1,1,1,2,1,1,2] => ? = 24
[3,6,9,2,5,8,1,4,7] => [7,4,1,8,5,2,9,6,3] => [1,1,2,1,2,1,1] => ? = 27
[3,6,2,5,1,4,7,8,9] => [5,3,1,6,4,2,7,8,9] => [1,1,2,1,4] => ? = 12
[2,4,6,8,1,3,5,7,9] => [5,1,6,2,7,3,8,4,9] => [1,2,2,2,2] => ? = 16
[2,4,6,1,3,5,7,8,9] => [4,1,5,2,6,3,7,8,9] => [1,2,2,4] => ? = 9
[5,10,4,9,3,8,2,7,1,6] => [9,7,5,3,1,10,8,6,4,2] => [1,1,1,1,2,1,1,1,1] => ? = 40
[4,8,3,7,2,6,1,5,9,10] => [7,5,3,1,8,6,4,2,9,10] => [1,1,1,2,1,1,3] => ? = 24
[3,6,9,2,5,8,1,4,7,10] => [7,4,1,8,5,2,9,6,3,10] => [1,1,2,1,2,1,2] => ? = 27
[3,6,2,5,1,4,7,8,9,10] => [5,3,1,6,4,2,7,8,9,10] => [1,1,2,1,5] => ? = 12
[2,4,6,8,1,3,5,7,9,10] => [5,1,6,2,7,3,8,4,9,10] => [1,2,2,2,3] => ? = 16
[2,4,6,1,3,5,7,8,9,10] => [4,1,5,2,6,3,7,8,9,10] => [1,2,2,5] => ? = 9
[2,4,1,3,5,6,7,8,9,10] => [3,1,4,2,5,6,7,8,9,10] => [1,2,7] => ? = 4
[8,7,6,5,4,2,1,3,9] => [7,6,8,5,4,3,2,1,9] => [1,2,1,1,1,1,2] => ? = 26
[1,2,9,8,7,6,5,4,3] => [1,2,9,8,7,6,5,4,3] => [3,1,1,1,1,1,1] => ? = 33
[1,2,3,9,8,7,6,5,4] => [1,2,3,9,8,7,6,5,4] => [4,1,1,1,1,1] => ? = 30
Description
The major index of the composition. The descents of a composition [c1,c2,,ck] are the partial sums c1,c1+c2,,c1++ck1, 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]].
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
Mp00085: Standard tableaux Schützenberger involutionStandard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 91%
Values
[1] => [[1]]
=> [[1]]
=> 0
[1,2] => [[1,2]]
=> [[1,2]]
=> 0
[2,1] => [[1],[2]]
=> [[1],[2]]
=> 1
[1,2,3] => [[1,2,3]]
=> [[1,2,3]]
=> 0
[1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,1,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[3,2,1] => [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[1,2,3,4] => [[1,2,3,4]]
=> [[1,2,3,4]]
=> 0
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[2,3,1,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 4
[2,4,3,1] => [[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 4
[3,1,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [[1,2],[3,4]]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[3,2,4,1] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,2],[3,4]]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 3
[4,1,2,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 3
[4,1,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[4,2,1,3] => [[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 4
[4,2,3,1] => [[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 4
[4,3,1,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 5
[4,3,2,1] => [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 6
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 4
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 7
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 2
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 6
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 6
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 3
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> 5
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> 5
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 3
[7,8,4,3,5,6,2,1] => ?
=> ?
=> ? = 12
[7,4,3,5,6,1,2,8] => ?
=> ?
=> ? = 11
[2,4,3,5,8,7,6,1] => ?
=> ?
=> ? = 17
[2,4,5,7,6,3,8,1] => ?
=> ?
=> ? = 10
[4,6,5,3,2,7,8,1] => ?
=> ?
=> ? = 11
[1,3,5,4,6,7,8,2] => ?
=> ?
=> ? = 6
[1,2,4,6,5,7,8,3] => ?
=> ?
=> ? = 8
[4,5,3,2,1,7,8,6] => ?
=> ?
=> ? = 12
[2,4,3,5,1,6,8,7] => ?
=> ?
=> ? = 11
[2,4,3,5,6,7,1,8] => ?
=> ?
=> ? = 4
[1,3,4,6,5,7,2,8] => ?
=> ?
=> ? = 7
[2,1,3,4,6,7,5,8] => ?
=> ?
=> ? = 6
[2,3,5,4,6,1,7,8] => ?
=> ?
=> ? = 5
[2,1,3,5,6,4,7,8] => ?
=> ?
=> ? = 5
[1,2,4,5,3,6,7,8] => ?
=> ?
=> ? = 3
[2,5,3,4,8,7,6,1] => ?
=> ?
=> ? = 18
[2,4,7,1,3,5,8,6] => ?
=> ?
=> ? = 10
[2,4,5,1,6,7,3,8] => ?
=> ?
=> ? = 4
[2,5,6,1,7,3,4,8] => ?
=> ?
=> ? = 5
[2,1,3,5,7,8,4,6] => ?
=> ?
=> ? = 11
[2,3,6,7,1,4,5,8] => ?
=> ?
=> ? = 6
[2,3,4,6,7,1,8,5] => ?
=> ?
=> ? = 6
[3,4,7,1,2,8,5,6] => ?
=> ?
=> ? = 8
[3,5,7,1,2,8,4,6] => ?
=> ?
=> ? = 12
[3,1,5,2,4,6,7,8] => ?
=> ?
=> ? = 6
[1,3,5,2,6,7,8,4] => ?
=> ?
=> ? = 6
[3,1,6,7,8,2,4,5] => ?
=> ?
=> ? = 7
[1,4,5,8,2,3,6,7] => ?
=> ?
=> ? = 10
[4,6,1,7,2,3,8,5] => ?
=> ?
=> ? = 8
[1,2,4,6,7,3,5,8] => ?
=> ?
=> ? = 8
[1,2,4,6,3,7,8,5] => ?
=> ?
=> ? = 8
[1,2,4,6,7,3,8,5] => ?
=> ?
=> ? = 8
[5,6,1,2,7,3,4,8] => ?
=> ?
=> ? = 4
[1,5,6,2,3,7,4,8] => ?
=> ?
=> ? = 4
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 35
[4,3,5,7,1,2,6,8] => ?
=> ?
=> ? = 11
[7,2,4,6,1,3,5,8] => ?
=> ?
=> ? = 15
[3,6,2,5,1,4,7,8] => ?
=> ?
=> ? = 12
[6,8,2,5,1,3,4,7] => ?
=> ?
=> ? = 17
[5,8,4,7,1,2,3,6] => ?
=> ?
=> ? = 20
[9,3,1,2,4,5,6,7,8] => [[1,2,4,5,6,7,8],[3],[9]]
=> [[1,3,4,5,6,7,9],[2],[8]]
=> ? = 10
[3,6,8,1,2,7,5,4] => ?
=> ?
=> ? = 18
[2,4,5,8,7,1,6,3] => ?
=> ?
=> ? = 17
[3,6,2,7,1,4,8,5] => ?
=> ?
=> ? = 8
[6,8,5,3,7,1,2,4] => ?
=> ?
=> ? = 18
[4,6,3,7,1,8,2,5] => ?
=> ?
=> ? = 10
[7,5,3,8,1,6,2,4] => ?
=> ?
=> ? = 12
[12,11,10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 66
[6,8,2,4,1,5,7,3] => ?
=> ?
=> ? = 16
[1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 10
Description
The cocharge of a standard tableau. The '''cocharge''' of a standard tableau T, denoted cc(T), is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation w1w2wn can be computed by the following algorithm: 1) Starting from wn, 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.
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 85% values known / values provided: 85%distinct values known / distinct values provided: 91%
Values
[1] => [[1]]
=> 0
[1,2] => [[1,2]]
=> 0
[2,1] => [[1],[2]]
=> 1
[1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [[1,2],[3]]
=> 2
[2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [[1,3],[2]]
=> 1
[3,1,2] => [[1,2],[3]]
=> 2
[3,2,1] => [[1],[2],[3]]
=> 3
[1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,2,4] => [[1,2,4],[3]]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> 3
[1,4,3,2] => [[1,2],[3],[4]]
=> 5
[2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [[1,3],[2,4]]
=> 4
[2,3,1,4] => [[1,3,4],[2]]
=> 1
[2,3,4,1] => [[1,3,4],[2]]
=> 1
[2,4,1,3] => [[1,3],[2,4]]
=> 4
[2,4,3,1] => [[1,3],[2],[4]]
=> 4
[3,1,2,4] => [[1,2,4],[3]]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> 3
[3,2,4,1] => [[1,4],[2],[3]]
=> 3
[3,4,1,2] => [[1,2],[3,4]]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> 3
[4,1,2,3] => [[1,2,3],[4]]
=> 3
[4,1,3,2] => [[1,2],[3],[4]]
=> 5
[4,2,1,3] => [[1,3],[2],[4]]
=> 4
[4,2,3,1] => [[1,3],[2],[4]]
=> 4
[4,3,1,2] => [[1,2],[3],[4]]
=> 5
[4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> 3
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> 3
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 7
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> 2
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> 2
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> 6
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> 6
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> 3
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 5
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> 5
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> 3
[7,8,4,3,5,6,2,1] => ?
=> ? = 12
[7,4,3,5,6,1,2,8] => ?
=> ? = 11
[2,4,3,5,8,7,6,1] => ?
=> ? = 17
[2,4,5,7,6,3,8,1] => ?
=> ? = 10
[4,6,5,3,2,7,8,1] => ?
=> ? = 11
[1,3,5,4,6,7,8,2] => ?
=> ? = 6
[1,2,4,6,5,7,8,3] => ?
=> ? = 8
[4,5,3,2,1,7,8,6] => ?
=> ? = 12
[2,4,3,5,1,6,8,7] => ?
=> ? = 11
[2,4,3,5,6,7,1,8] => ?
=> ? = 4
[1,3,4,6,5,7,2,8] => ?
=> ? = 7
[2,1,3,4,6,7,5,8] => ?
=> ? = 6
[2,3,5,4,6,1,7,8] => ?
=> ? = 5
[2,1,3,5,6,4,7,8] => ?
=> ? = 5
[1,2,4,5,3,6,7,8] => ?
=> ? = 3
[2,5,3,4,8,7,6,1] => ?
=> ? = 18
[2,4,7,1,3,5,8,6] => ?
=> ? = 10
[2,4,5,1,6,7,3,8] => ?
=> ? = 4
[2,5,6,1,7,3,4,8] => ?
=> ? = 5
[2,1,3,5,7,8,4,6] => ?
=> ? = 11
[2,3,6,7,1,4,5,8] => ?
=> ? = 6
[2,3,4,6,7,1,8,5] => ?
=> ? = 6
[3,4,7,1,2,8,5,6] => ?
=> ? = 8
[3,5,7,1,2,8,4,6] => ?
=> ? = 12
[3,1,5,2,4,6,7,8] => ?
=> ? = 6
[1,3,5,2,6,7,8,4] => ?
=> ? = 6
[3,1,6,7,8,2,4,5] => ?
=> ? = 7
[1,4,5,8,2,3,6,7] => ?
=> ? = 10
[4,6,1,7,2,3,8,5] => ?
=> ? = 8
[1,2,4,6,7,3,5,8] => ?
=> ? = 8
[1,2,4,6,3,7,8,5] => ?
=> ? = 8
[1,2,4,6,7,3,8,5] => ?
=> ? = 8
[5,6,1,2,7,3,4,8] => ?
=> ? = 4
[1,5,6,2,3,7,4,8] => ?
=> ? = 4
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 35
[4,3,5,7,1,2,6,8] => ?
=> ? = 11
[7,2,4,6,1,3,5,8] => ?
=> ? = 15
[3,6,2,5,1,4,7,8] => ?
=> ? = 12
[6,8,2,5,1,3,4,7] => ?
=> ? = 17
[5,8,4,7,1,2,3,6] => ?
=> ? = 20
[9,3,1,2,4,5,6,7,8] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 10
[3,6,8,1,2,7,5,4] => ?
=> ? = 18
[2,4,5,8,7,1,6,3] => ?
=> ? = 17
[3,6,2,7,1,4,8,5] => ?
=> ? = 8
[6,8,5,3,7,1,2,4] => ?
=> ? = 18
[4,6,3,7,1,8,2,5] => ?
=> ? = 10
[7,5,3,8,1,6,2,4] => ?
=> ? = 12
[12,11,10,9,8,7,6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 66
[6,8,2,4,1,5,7,3] => ?
=> ? = 16
[1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
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.
Mp00066: Permutations inversePermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St001161: Dyck paths ⟶ ℤResult quality: 67% values known / values provided: 85%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,4,3,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,4,3,1] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,2,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[4,2,1,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[4,2,3,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[4,3,1,2] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[7,8,6,5,4,1,2,3] => [6,7,8,5,4,3,1,2] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[6,7,8,5,4,1,2,3] => [6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[7,6,5,4,1,2,3,8] => [5,6,7,4,3,2,1,8] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[7,1,2,3,4,5,6,8] => [2,3,4,5,6,7,1,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[6,1,2,3,4,5,7,8] => [2,3,4,5,6,1,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[5,1,2,3,4,6,7,8] => [2,3,4,5,1,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,2,8,7,6,5,4,3] => [1,2,8,7,6,5,4,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,2,6,7,8,5,4,3] => [1,2,8,7,6,3,4,5] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[1,2,5,6,8,7,4,3] => [1,2,8,7,3,4,6,5] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 14
[1,2,5,6,4,7,8,3] => [1,2,8,5,3,4,6,7] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,2,4,6,5,7,8,3] => [1,2,8,3,5,4,6,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 22
[1,2,3,5,7,8,6,4] => [1,2,3,8,4,7,5,6] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[1,2,3,6,7,5,8,4] => [1,2,3,8,6,4,5,7] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,2,3,5,6,7,8,4] => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,4,8,7,6,5] => [1,2,3,4,8,7,6,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 18
[1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 13
[1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,4,6,7,5,3,8] => [1,2,7,3,6,4,5,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,4,5,6,7,3,8] => [1,2,7,3,4,5,6,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,2,3,6,5,7,4,8] => [1,2,3,7,5,4,6,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,2,4,3,6,7,5,8] => [1,2,4,3,7,5,6,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,3,4,6,7,5,8] => [1,2,3,4,7,5,6,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,2,3,5,6,4,7,8] => [1,2,3,6,4,5,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,4,6,5,7,8] => [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,4,5,3,6,7,8] => [1,2,5,3,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,7,5,6,8,3] => [1,2,8,3,5,6,4,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[4,5,1,6,2,7,3,8] => [3,5,7,1,2,4,6,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[4,5,7,1,2,8,3,6] => [4,5,7,1,2,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[4,1,2,5,3,7,8,6] => [2,3,5,1,4,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,4,5,2,3,7,8,6] => [1,4,5,2,3,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,4,5,7,8,2,3,6] => [1,6,7,2,3,8,4,5] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,4,5,8,2,3,6,7] => [1,5,6,2,3,7,8,4] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,4,5,2,3,6,7,8] => [1,4,5,2,3,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,4,5,2,6,7,3,8] => [1,4,7,2,3,5,6,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,4,5,6,7,2,3,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[4,6,1,7,2,3,8,5] => [3,5,6,1,8,2,4,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[4,6,1,2,7,8,3,5] => [3,4,7,1,8,2,5,6] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[4,6,7,1,8,2,3,5] => [4,6,7,1,8,2,3,5] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[4,6,7,1,2,3,5,8] => [4,5,6,1,7,2,3,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,4,6,7,2,8,3,5] => [1,5,7,2,8,3,4,6] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
Description
The major index north count of a Dyck path. The descent set des(D) of a Dyck path D=D1D2n with Di{N,E} is given by all indices i such that Di=E and Di+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, ides(D)i, see [[St000027]]. The '''major index north count''' is given by ides(D)#{jiDj=N}.
Mp00066: Permutations inversePermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
St000947: Dyck paths ⟶ ℤResult quality: 67% values known / values provided: 85%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,4,3,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[2,4,3,1] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,2,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[4,2,1,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[4,2,3,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[4,3,1,2] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,4,5,3,2] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[7,8,6,5,4,1,2,3] => [6,7,8,5,4,3,1,2] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[6,7,8,5,4,1,2,3] => [6,7,8,5,4,1,2,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[7,6,5,4,1,2,3,8] => [5,6,7,4,3,2,1,8] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[7,1,2,3,4,5,6,8] => [2,3,4,5,6,7,1,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[6,1,2,3,4,5,7,8] => [2,3,4,5,6,1,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[5,1,2,3,4,6,7,8] => [2,3,4,5,1,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,2,8,7,6,5,4,3] => [1,2,8,7,6,5,4,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 25
[1,2,6,7,8,5,4,3] => [1,2,8,7,6,3,4,5] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[1,2,5,6,8,7,4,3] => [1,2,8,7,3,4,6,5] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 14
[1,2,5,6,4,7,8,3] => [1,2,8,5,3,4,6,7] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,2,4,6,5,7,8,3] => [1,2,8,3,5,4,6,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 22
[1,2,3,5,7,8,6,4] => [1,2,3,8,4,7,5,6] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[1,2,3,6,7,5,8,4] => [1,2,3,8,6,4,5,7] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,2,3,5,6,7,8,4] => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,4,8,7,6,5] => [1,2,3,4,8,7,6,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 18
[1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 13
[1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,2,4,6,7,5,3,8] => [1,2,7,3,6,4,5,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,4,5,6,7,3,8] => [1,2,7,3,4,5,6,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,2,3,6,5,7,4,8] => [1,2,3,7,5,4,6,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 9
[1,2,4,3,6,7,5,8] => [1,2,4,3,7,5,6,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[1,2,3,4,6,7,5,8] => [1,2,3,4,7,5,6,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,2,3,5,6,4,7,8] => [1,2,3,6,4,5,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,3,4,6,5,7,8] => [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,2,4,5,3,6,7,8] => [1,2,5,3,4,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,7,5,6,8,3] => [1,2,8,3,5,6,4,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[4,5,1,6,2,7,3,8] => [3,5,7,1,2,4,6,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[4,5,7,1,2,8,3,6] => [4,5,7,1,2,8,3,6] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[4,1,2,5,3,7,8,6] => [2,3,5,1,4,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,4,5,2,3,7,8,6] => [1,4,5,2,3,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,4,5,7,8,2,3,6] => [1,6,7,2,3,8,4,5] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,4,5,8,2,3,6,7] => [1,5,6,2,3,7,8,4] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,4,5,2,3,6,7,8] => [1,4,5,2,3,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,4,5,2,6,7,3,8] => [1,4,7,2,3,5,6,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,4,5,6,7,2,3,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[4,6,1,7,2,3,8,5] => [3,5,6,1,8,2,4,7] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[4,6,1,2,7,8,3,5] => [3,4,7,1,8,2,5,6] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[4,6,7,1,8,2,3,5] => [4,6,7,1,8,2,3,5] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[4,6,7,1,2,3,5,8] => [4,5,6,1,7,2,3,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
Description
The major index east count of a Dyck path. The descent set des(D) of a Dyck path D=D1D2n with Di{N,E} is given by all indices i such that Di=E and Di+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, ides(D)i, see [[St000027]]. The '''major index east count''' is given by ides(D)#{jiDj=E}.
Mp00069: Permutations complementPermutations
Mp00059: Permutations Robinson-Schensted insertion tableauStandard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 84% values known / values provided: 84%distinct values known / distinct values provided: 95%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [2,1] => [[1],[2]]
=> 0
[2,1] => [1,2] => [[1,2]]
=> 1
[1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 0
[1,3,2] => [3,1,2] => [[1,2],[3]]
=> 2
[2,1,3] => [2,3,1] => [[1,3],[2]]
=> 1
[2,3,1] => [2,1,3] => [[1,3],[2]]
=> 1
[3,1,2] => [1,3,2] => [[1,2],[3]]
=> 2
[3,2,1] => [1,2,3] => [[1,2,3]]
=> 3
[1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,2,4,3] => [4,3,1,2] => [[1,2],[3],[4]]
=> 3
[1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[1,4,2,3] => [4,1,3,2] => [[1,2],[3],[4]]
=> 3
[1,4,3,2] => [4,1,2,3] => [[1,2,3],[4]]
=> 5
[2,1,3,4] => [3,4,2,1] => [[1,4],[2],[3]]
=> 1
[2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 4
[2,3,1,4] => [3,2,4,1] => [[1,4],[2],[3]]
=> 1
[2,3,4,1] => [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[2,4,1,3] => [3,1,4,2] => [[1,2],[3,4]]
=> 4
[2,4,3,1] => [3,1,2,4] => [[1,2,4],[3]]
=> 4
[3,1,2,4] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
[3,1,4,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [2,3,4,1] => [[1,3,4],[2]]
=> 3
[3,2,4,1] => [2,3,1,4] => [[1,3,4],[2]]
=> 3
[3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 3
[4,1,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 5
[4,2,1,3] => [1,3,4,2] => [[1,2,4],[3]]
=> 4
[4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 4
[4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 5
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 6
[1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 4
[1,2,4,3,5] => [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,2,4,5,3] => [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 3
[1,2,5,3,4] => [5,4,1,3,2] => [[1,2],[3],[4],[5]]
=> 4
[1,2,5,4,3] => [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 7
[1,3,2,4,5] => [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 2
[1,3,2,5,4] => [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 6
[1,3,4,2,5] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 2
[1,3,4,5,2] => [5,3,2,1,4] => [[1,4],[2],[3],[5]]
=> 2
[1,3,5,2,4] => [5,3,1,4,2] => [[1,2],[3,4],[5]]
=> 6
[1,3,5,4,2] => [5,3,1,2,4] => [[1,2,4],[3],[5]]
=> 6
[1,4,2,3,5] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 3
[1,4,2,5,3] => [5,2,4,1,3] => [[1,3],[2,4],[5]]
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4,3,5,2] => [5,2,3,1,4] => [[1,3,4],[2],[5]]
=> 5
[1,4,5,2,3] => [5,2,1,4,3] => [[1,3],[2,4],[5]]
=> 3
[6,5,7,8,4,3,1,2] => [3,4,2,1,5,6,8,7] => ?
=> ? = 14
[7,4,3,5,6,1,2,8] => [2,5,6,4,3,8,7,1] => ?
=> ? = 11
[3,5,4,6,7,8,2,1] => [6,4,5,3,2,1,7,8] => ?
=> ? = 7
[3,4,5,2,7,8,6,1] => [6,5,4,7,2,1,3,8] => ?
=> ? = 9
[2,4,3,5,8,7,6,1] => [7,5,6,4,1,2,3,8] => ?
=> ? = 17
[2,4,5,7,6,3,8,1] => [7,5,4,2,3,6,1,8] => ?
=> ? = 10
[2,3,5,6,7,4,8,1] => [7,6,4,3,2,5,1,8] => ?
=> ? = 5
[4,6,5,3,2,7,8,1] => [5,3,4,6,7,2,1,8] => ?
=> ? = 11
[3,5,4,6,2,7,8,1] => [6,4,5,3,7,2,1,8] => ?
=> ? = 7
[1,2,5,6,8,7,4,3] => [8,7,4,3,1,2,5,6] => ?
=> ? = 14
[1,2,5,6,4,7,8,3] => [8,7,4,3,5,2,1,6] => ?
=> ? = 7
[2,5,4,3,1,8,7,6] => [7,4,5,6,8,1,2,3] => ?
=> ? = 21
[1,3,4,5,2,7,8,6] => [8,6,5,4,7,2,1,3] => ?
=> ? = 8
[2,4,3,5,6,7,1,8] => [7,5,6,4,3,2,8,1] => ?
=> ? = 4
[1,3,4,6,5,7,2,8] => [8,6,5,3,4,2,7,1] => ?
=> ? = 7
[1,4,5,3,6,7,2,8] => [8,5,4,6,3,2,7,1] => ?
=> ? = 5
[1,2,4,5,6,7,3,8] => [8,7,5,4,3,2,6,1] => ?
=> ? = 3
[2,3,5,4,6,1,7,8] => [7,6,4,5,3,8,2,1] => ?
=> ? = 5
[1,2,4,7,5,6,8,3] => [8,7,5,2,4,3,1,6] => ?
=> ? = 9
[1,6,4,3,5,8,7,2] => [8,3,5,6,4,1,2,7] => ?
=> ? = 17
[2,1,4,8,6,5,7,3] => [7,8,5,1,3,4,2,6] => ?
=> ? = 16
[2,5,3,4,8,7,6,1] => [7,4,6,5,1,2,3,8] => ?
=> ? = 18
[2,8,4,5,6,3,7,1] => [7,1,5,4,3,6,2,8] => ?
=> ? = 11
[2,4,1,5,3,7,8,6] => [7,5,8,4,6,2,1,3] => ?
=> ? = 10
[2,4,5,1,6,3,8,7] => [7,5,4,8,3,6,1,2] => ?
=> ? = 11
[2,4,5,6,1,3,7,8] => [7,5,4,3,8,6,2,1] => ?
=> ? = 4
[2,4,5,1,6,7,3,8] => [7,5,4,8,3,2,6,1] => ?
=> ? = 4
[2,4,5,1,6,7,8,3] => [7,5,4,8,3,2,1,6] => ?
=> ? = 4
[2,1,5,6,7,3,8,4] => [7,8,4,3,2,6,1,5] => ?
=> ? = 5
[2,1,3,5,7,8,4,6] => [7,8,6,4,2,1,5,3] => ?
=> ? = 11
[2,3,4,6,7,1,8,5] => [7,6,5,3,2,8,1,4] => ?
=> ? = 6
[2,3,7,1,4,5,8,6] => [7,6,2,8,5,4,1,3] => ?
=> ? = 7
[3,4,1,5,2,6,7,8] => [6,5,8,4,7,3,2,1] => ?
=> ? = 2
[3,4,1,5,6,7,2,8] => [6,5,8,4,3,2,7,1] => ?
=> ? = 2
[3,5,1,7,8,2,4,6] => [6,4,8,2,1,7,5,3] => ?
=> ? = 12
[3,1,5,2,4,6,7,8] => [6,8,4,7,5,3,2,1] => ?
=> ? = 6
[1,3,5,2,6,7,8,4] => [8,6,4,7,3,2,1,5] => ?
=> ? = 6
[3,1,6,7,8,2,4,5] => [6,8,3,2,1,7,5,4] => ?
=> ? = 7
[1,3,4,6,7,2,5,8] => [8,6,5,3,2,7,4,1] => ?
=> ? = 7
[1,4,5,2,6,7,3,8] => [8,5,4,7,3,2,6,1] => ?
=> ? = 3
[1,2,4,6,3,7,8,5] => [8,7,5,3,6,2,1,4] => ?
=> ? = 8
[1,2,4,5,7,3,6,8] => [8,7,5,4,2,6,3,1] => ?
=> ? = 9
[1,2,4,5,7,8,3,6] => [8,7,5,4,2,1,6,3] => ?
=> ? = 9
[1,5,6,2,3,7,4,8] => [8,4,3,7,6,2,5,1] => ?
=> ? = 4
[1,2,3,5,7,4,6,8] => [8,7,6,4,2,5,3,1] => ?
=> ? = 10
[9,10,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,10,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 35
[4,3,5,7,1,2,6,8] => [5,6,4,2,8,7,3,1] => ?
=> ? = 11
[6,1,5,2,3,7,8,4] => [3,8,4,7,6,2,1,5] => ?
=> ? = 9
[8,3,1,6,2,4,5,7] => [1,6,8,3,7,5,4,2] => ?
=> ? = 14
[9,3,1,2,4,5,6,7,8] => [1,7,9,8,6,5,4,3,2] => [[1,2,8],[3],[4],[5],[6],[7],[9]]
=> ? = 10
Description
The charge of a standard tableau.
Mp00064: Permutations reversePermutations
Mp00066: Permutations inversePermutations
Mp00067: Permutations Foata bijectionPermutations
St000246: Permutations ⟶ ℤResult quality: 57% values known / values provided: 57%distinct values known / distinct values provided: 84%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,1,2] => [1,3,2] => 2
[2,1,3] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [1,4,3,2] => 3
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [2,4,3,1] => 2
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 2
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [2,4,1,3] => 3
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 5
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,3,4,2] => 4
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [4,2,3,1] => 1
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 1
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,1,4,3] => 4
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 4
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 3
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [3,1,4,2] => 3
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 5
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 4
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 4
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 5
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 6
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [1,5,4,3,2] => 4
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [2,5,4,3,1] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [5,1,4,3,2] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [2,5,4,1,3] => 4
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [1,2,5,4,3] => 7
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [3,5,4,2,1] => 2
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [1,3,5,4,2] => 6
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [5,4,1,3,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [2,1,5,4,3] => 6
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [1,5,2,4,3] => 6
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [3,5,2,4,1] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [3,5,4,1,2] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [2,3,5,4,1] => 5
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [3,1,5,4,2] => 5
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [5,2,4,1,3] => 3
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,6,5,4,2,3,1] => [2,7,6,5,4,3,1] => ? = 5
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [7,6,5,4,1,3,2] => [7,1,6,5,4,3,2] => ? = 5
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => [2,7,6,5,4,1,3] => ? = 6
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => ? = 11
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,5,3,4,2,1] => [3,7,6,5,4,2,1] => ? = 4
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,6,5,3,2,4,1] => [3,7,6,5,2,4,1] => ? = 5
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [7,6,5,3,1,4,2] => [3,7,6,5,4,1,2] => ? = 5
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [7,6,5,2,1,4,3] => [7,2,6,5,4,1,3] => ? = 5
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => [3,7,6,5,2,1,4] => ? = 6
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? = 15
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,4,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 3
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,6,3,5,2,4,1] => [3,2,7,6,5,4,1] => ? = 8
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [7,6,2,5,1,4,3] => [7,2,1,6,5,4,3] => ? = 8
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [7,6,3,5,2,1,4] => [3,2,7,6,5,1,4] => ? = 9
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [7,6,3,5,1,2,4] => [1,3,2,7,6,5,4] => ? = 14
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [7,6,4,1,5,3,2] => [4,7,6,5,3,1,2] => ? = 4
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,6,2,4,5,3,1] => [4,2,7,6,5,3,1] => ? = 7
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => [7,6,3,1,5,4,2] => [7,3,6,5,4,1,2] => ? = 4
[1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [7,6,1,3,5,4,2] => [7,3,1,6,5,4,2] => ? = 7
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [7,6,2,1,5,4,3] => [7,6,2,5,4,1,3] => ? = 4
[1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => [7,6,2,4,1,5,3] => [4,2,7,6,5,1,3] => ? = 8
[1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? = 18
[1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [7,5,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 2
[1,3,2,5,6,4,7] => [7,4,6,5,2,3,1] => [7,5,6,2,4,3,1] => [5,2,7,6,4,3,1] => ? = 6
[1,3,4,5,7,2,6] => [6,2,7,5,4,3,1] => [7,2,6,5,4,1,3] => [2,7,6,1,5,4,3] => ? = 8
[1,3,4,6,2,5,7] => [7,5,2,6,4,3,1] => [7,3,6,5,2,4,1] => [3,7,2,6,5,4,1] => ? = 7
[1,3,4,6,7,2,5] => [5,2,7,6,4,3,1] => [7,2,6,5,1,4,3] => [7,2,6,1,5,4,3] => ? = 7
[1,3,4,7,2,5,6] => [6,5,2,7,4,3,1] => [7,3,6,5,2,1,4] => [3,7,2,6,5,1,4] => ? = 8
[1,3,5,2,6,4,7] => [7,4,6,2,5,3,1] => [7,4,6,2,5,3,1] => [4,7,2,6,5,3,1] => ? = 6
[1,3,5,4,6,7,2] => [2,7,6,4,5,3,1] => [7,1,6,4,5,3,2] => [7,4,1,6,5,3,2] => ? = 6
[1,3,5,6,2,7,4] => [4,7,2,6,5,3,1] => [7,3,6,1,5,4,2] => [7,3,6,1,5,4,2] => ? = 6
[1,3,5,7,2,4,6] => [6,4,2,7,5,3,1] => [7,3,6,2,5,1,4] => [3,2,1,7,6,5,4] => ? = 12
[1,3,7,2,6,5,4] => [4,5,6,2,7,3,1] => [7,4,6,1,2,3,5] => [1,2,4,3,7,6,5] => ? = 17
[1,4,2,5,6,7,3] => [3,7,6,5,2,4,1] => [7,5,1,6,4,3,2] => [5,7,6,4,3,1,2] => ? = 3
[1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => [7,2,1,6,5,4,3] => [7,6,5,2,4,1,3] => ? = 3
[1,4,6,2,3,5,7] => [7,5,3,2,6,4,1] => [7,4,3,6,2,5,1] => [4,3,7,2,6,5,1] => ? = 8
[1,4,6,7,2,3,5] => [5,3,2,7,6,4,1] => [7,3,2,6,1,5,4] => [7,3,2,6,1,5,4] => ? = 8
[1,4,7,3,6,2,5] => [5,2,6,3,7,4,1] => [7,2,4,6,1,3,5] => [2,1,4,3,7,6,5] => ? = 16
[2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 1
[2,1,3,4,6,7,5] => [5,7,6,4,3,1,2] => [6,7,5,4,1,3,2] => [6,1,7,5,4,3,2] => ? = 6
[2,1,3,5,6,4,7] => [7,4,6,5,3,1,2] => [6,7,5,2,4,3,1] => [6,2,7,5,4,3,1] => ? = 5
[2,1,3,5,6,7,4] => [4,7,6,5,3,1,2] => [6,7,5,1,4,3,2] => [6,7,1,5,4,3,2] => ? = 5
[2,1,4,5,3,6,7] => [7,6,3,5,4,1,2] => [6,7,3,5,4,2,1] => [6,3,7,5,4,2,1] => ? = 4
[2,1,4,5,6,3,7] => [7,3,6,5,4,1,2] => [6,7,2,5,4,3,1] => [6,7,2,5,4,3,1] => ? = 4
[2,1,4,5,6,7,3] => [3,7,6,5,4,1,2] => [6,7,1,5,4,3,2] => [6,7,5,1,4,3,2] => ? = 4
[2,1,4,6,7,3,5] => [5,3,7,6,4,1,2] => [6,7,2,5,1,4,3] => [6,2,1,7,5,4,3] => ? = 9
[2,1,5,6,4,7,3] => [3,7,4,6,5,1,2] => [6,7,1,3,5,4,2] => [6,3,1,7,5,4,2] => ? = 8
[2,3,1,4,5,6,7] => [7,6,5,4,1,3,2] => [5,7,6,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 1
[2,3,1,5,6,7,4] => [4,7,6,5,1,3,2] => [5,7,6,1,4,3,2] => [7,5,1,6,4,3,2] => ? = 5
[2,3,4,5,6,1,7] => [7,1,6,5,4,3,2] => [2,7,6,5,4,3,1] => [7,6,5,4,2,3,1] => ? = 1
Description
The number of non-inversions of a permutation. For a permutation of {1,,n}, this is given by \operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi).
Matching statistic: St000579
Mp00072: Permutations binary search tree: left to rightBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000579: Set partitions ⟶ ℤResult quality: 51% values known / values provided: 51%distinct values known / distinct values provided: 51%
Values
[1] => [.,.]
=> [1,0]
=> {{1}}
=> ? = 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> {{1},{2}}
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 5
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 4
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 5
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 4
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 5
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 6
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 7
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 6
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 6
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 6
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 28
[7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 21
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 18
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 20
[7,8,4,3,5,6,2,1] => ?
=> ?
=> ?
=> ? = 12
[7,8,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 19
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 20
[8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5},{6,7},{8}}
=> ? = 19
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 12
[5,4,6,3,7,2,8,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 10
[6,4,3,5,7,2,8,1] => [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 11
[7,2,3,4,5,6,8,1] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 7
[5,3,4,2,6,7,8,1] => [[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2},{3,4},{5,6,7,8}}
=> ? = 7
[4,3,2,5,6,7,8,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 6
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 20
[6,5,7,8,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4},{5},{6,7,8}}
=> ? = 14
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 12
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2},{3},{4,5,6},{7,8}}
=> ? = 11
[7,3,4,5,6,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 8
[8,7,6,5,4,2,1,3] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 26
[7,8,6,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 19
[7,6,8,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 19
[7,8,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5},{6},{7,8}}
=> ? = 18
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 12
[7,8,6,5,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 18
[7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 18
[6,5,7,8,3,2,1,4] => [[[[[.,.],.],[.,.]],.],[.,[.,.]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 12
[7,5,6,8,2,3,1,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6},{7,8}}
=> ? = 11
[6,7,5,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3,4},{5},{6,7,8}}
=> ? = 11
[6,5,7,8,2,1,3,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 10
[5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4
[8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 24
[8,7,6,4,2,1,3,5] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 22
[8,6,5,4,3,2,1,7] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 22
[8,6,5,4,2,1,3,7] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 20
[8,6,4,3,2,1,5,7] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 18
[8,6,4,2,1,3,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 16
[8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3},{4,5,6,7},{8}}
=> ? = 10
[8,3,2,1,4,5,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1},{2},{3,4,5,6,7},{8}}
=> ? = 10
[8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 9
[7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 21
[7,6,5,3,4,2,1,8] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 18
[7,6,5,4,2,3,1,8] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 19
[7,6,5,4,3,1,2,8] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 20
[7,4,3,5,6,1,2,8] => ?
=> ?
=> ?
=> ? = 11
[7,6,5,4,2,1,3,8] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 19
[7,6,5,4,1,2,3,8] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5},{6},{7,8}}
=> ? = 18
[7,6,5,3,2,1,4,8] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 18
[6,5,4,3,2,1,7,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 15
Description
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. This is the number of pairs i\lt j in different blocks such that j is the maximal element of a block.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000081The number of edges of a graph. St001759The Rajchgot index of a permutation. St001397Number of pairs of incomparable elements in a finite poset. St000446The disorder of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000797The stat`` of a permutation. St000795The mad of a permutation. St000004The major index of a permutation. St000305The inverse major index of a permutation. St000304The load of a permutation. St000005The bounce statistic of a Dyck path. St000154The sum of the descent bottoms 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. St001209The pmaj statistic of a parking function. St001931The weak major index of an integer composition regarded as a word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.