- FindStat

Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000008

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => [1] => 0
[[1,2]]
 => 0 => [2] => [1,1] => 1
[[1],[2]]
 => 1 => [1,1] => [2] => 0
[[1,2,3]]
 => 00 => [3] => [1,1,1] => 3
[[1,3],[2]]
 => 10 => [1,2] => [1,2] => 1
[[1,2],[3]]
 => 01 => [2,1] => [2,1] => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => [3] => 0
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1] => 6
[[1,3,4],[2]]
 => 100 => [1,3] => [1,1,2] => 3
[[1,2,4],[3]]
 => 010 => [2,2] => [1,2,1] => 4
[[1,2,3],[4]]
 => 001 => [3,1] => [2,1,1] => 5
[[1,3],[2,4]]
 => 101 => [1,2,1] => [2,2] => 2
[[1,2],[3,4]]
 => 010 => [2,2] => [1,2,1] => 4
[[1,4],[2],[3]]
 => 110 => [1,1,2] => [1,3] => 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [2,2] => 2
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [3,1] => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [4] => 0
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1] => 10
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,1,1,2] => 6
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,2,1] => 7
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,2,1,1] => 8
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [2,1,1,1] => 9
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,2,2] => 4
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,2,1] => 7
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [2,1,2] => 5
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [2,2,1] => 6
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,2,1,1] => 8
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,1,3] => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,2,2] => 4
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,3,1] => 5
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [2,1,2] => 5
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [2,2,1] => 6
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [3,1,1] => 7
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [2,3] => 2
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,2,2] => 4
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,3,1] => 5
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [3,2] => 3
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [2,2,1] => 6
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,4] => 1
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [2,3] => 2
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [3,2] => 3
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [4,1] => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [5] => 0
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1] => 15
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,1,1,1,2] => 10
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,1,2,1] => 11
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,2,1,1] => 12
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,2,1,1,1] => 13
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [2,1,1,1,1] => 14
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,1,2,2] => 7

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: St000391

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 90%●distinct values known / distinct values provided: 71%

Values

[[1]]
 => [[1]]
 =>  =>  => ? = 0
[[1,2]]
 => [[1],[2]]
 => 1 => 1 => 1
[[1],[2]]
 => [[1,2]]
 => 0 => 0 => 0
[[1,2,3]]
 => [[1],[2],[3]]
 => 11 => 11 => 3
[[1,3],[2]]
 => [[1,2],[3]]
 => 01 => 10 => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => 10 => 01 => 2
[[1],[2],[3]]
 => [[1,2,3]]
 => 00 => 00 => 0
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 111 => 111 => 6
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 011 => 110 => 3
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 101 => 101 => 4
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 110 => 011 => 5
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 010 => 010 => 2
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 101 => 101 => 4
[[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 001 => 100 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 010 => 010 => 2
[[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 100 => 001 => 3
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 000 => 000 => 0
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 10
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 0111 => 1110 => 6
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 1011 => 1101 => 7
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 1101 => 1011 => 8
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 1110 => 0111 => 9
[[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 0101 => 1010 => 4
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 1011 => 1101 => 7
[[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => 0110 => 0110 => 5
[[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 1010 => 0101 => 6
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 1101 => 1011 => 8
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 0011 => 1100 => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 0101 => 1010 => 4
[[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => 1001 => 1001 => 5
[[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => 0110 => 0110 => 5
[[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 1010 => 0101 => 6
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 1100 => 0011 => 7
[[1,4],[2,5],[3]]
 => [[1,2,3],[4,5]]
 => 0010 => 0100 => 2
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 0101 => 1010 => 4
[[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 1001 => 1001 => 5
[[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 0100 => 0010 => 3
[[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 1010 => 0101 => 6
[[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 0001 => 1000 => 1
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => 0010 => 0100 => 2
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => 0100 => 0010 => 3
[[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 1000 => 0001 => 4
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0000 => 0000 => 0
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 11111 => 11111 => 15
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 01111 => 11110 => 10
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => 10111 => 11101 => 11
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => 11011 => 11011 => 12
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => 11101 => 10111 => 13
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 11110 => 01111 => 14
[[1,3,5,6],[2,4]]
 => [[1,2],[3,4],[5],[6]]
 => 01011 => 11010 => 7
[[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 10111 => 11101 => 11
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => 01010101010 => 01010101010 => ? = 30
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => [[1,2],[3,4],[5,6],[7,10],[8,11],[9,12]]
 => 01010111011 => 11011101010 => ? = 36
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => [[1,2],[3,4],[5,8],[6,9],[7,10],[11,12]]
 => 01011101110 => 01110111010 => ? = 40
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
 => [[1,2],[3,4],[5,8],[6,10],[7,11],[9,12]]
 => 01011101011 => 11010111010 => ? = 38
[[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => [[1,2],[3,5],[4,6],[7,9],[8,10],[11,12]]
 => 01101110110 => 01101110110 => ? = 42
[[1,3,4,6,8,9],[2,5,7,10,11,12]]
 => [[1,2],[3,5],[4,7],[6,10],[8,11],[9,12]]
 => 01101011011 => 11011010110 => ? = 38
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
 => [[1,2],[3,6],[4,7],[5,8],[9,10],[11,12]]
 => 01110111010 => 01011101110 => ? = 44
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
 => [[1,2],[3,6],[4,8],[5,9],[7,10],[11,12]]
 => 01110101110 => 01110101110 => ? = 42
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
 => [[1,2],[3,6],[4,8],[5,10],[7,11],[9,12]]
 => 01110101011 => 11010101110 => ? = 40
[[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => [[1,2],[3,8],[4,9],[5,10],[6,11],[7,12]]
 => 01111101111 => 11110111110 => ? = 50
[[1,2,3,7,9,11],[4,5,6,8,10,12]]
 => [[1,4],[2,5],[3,6],[7,8],[9,10],[11,12]]
 => 11011101010 => 01010111011 => ? = 48
[[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => [[1,4],[2,5],[3,6],[7,10],[8,11],[9,12]]
 => 11011111011 => 11011111011 => ? = 54
[[1,2,3,6,8,11],[4,5,7,9,10,12]]
 => [[1,4],[2,5],[3,7],[6,9],[8,10],[11,12]]
 => 11011010110 => 01101011011 => ? = 46
[[1,2,3,5,9,11],[4,6,7,8,10,12]]
 => [[1,4],[2,6],[3,7],[5,8],[9,10],[11,12]]
 => 11010111010 => 01011101011 => ? = 46
[[1,2,3,5,7,11],[4,6,8,9,10,12]]
 => [[1,4],[2,6],[3,8],[5,9],[7,10],[11,12]]
 => 11010101110 => 01110101011 => ? = 44
[[1,2,3,5,7,9],[4,6,8,10,11,12]]
 => [[1,4],[2,6],[3,8],[5,10],[7,11],[9,12]]
 => 11010101011 => 11010101011 => ? = 42
[[1,2,3,5,6,7],[4,8,9,10,11,12]]
 => [[1,4],[2,8],[3,9],[5,10],[6,11],[7,12]]
 => 11011101111 => 11110111011 => ? = 52
[[1,2,3,4,5,11],[6,7,8,9,10,12]]
 => [[1,6],[2,7],[3,8],[4,9],[5,10],[11,12]]
 => 11110111110 => 01111101111 => ? = 58
[[1,2,3,4,5,9],[6,7,8,10,11,12]]
 => [[1,6],[2,7],[3,8],[4,10],[5,11],[9,12]]
 => 11110111011 => 11011101111 => ? = 56
[[1,2,3,4,5,7],[6,8,9,10,11,12]]
 => [[1,6],[2,8],[3,9],[4,10],[5,11],[7,12]]
 => 11110101111 => 11110101111 => ? = 54
[[1,2,3,4,5,6,7,8,9,10,11,12]]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => 11111111111 => 11111111111 => ? = 66
[[1,2,3,4,5],[6,7,8,9],[10,11,12]]
 => [[1,6,10],[2,7,11],[3,8,12],[4,9],[5]]
 => 11110111011 => 11011101111 => ? = 56
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
 => [[1,6,8,10,12],[2,7,9,11],[3],[4],[5]]
 => 11110101010 => 01010101111 => ? = 50
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
 => 11011101111 => 11110111011 => ? = 52
[[1,4,7,9],[2,5,8,10],[3,6]]
 => ?
 => ? => ? => ? = 13
[[1,3,5,8],[2,4,6,9],[7,10]]
 => ?
 => ? => ? => ? = 17
[[1,2],[3,6],[4,7],[5,8],[9,10]]
 => ?
 => ? => ? => ? = 15
[[1,3],[2,5],[4,8],[6,9],[7,10]]
 => ?
 => ? => ? => ? = 17
[[1,4],[2,5],[3,7],[6,9],[8,10]]
 => ?
 => ? => ? => ? = 13
[[1,3],[2,5],[4,7],[6,9],[8,11],[10,12]]
 => ?
 => ? => ? => ? = 30
[[1,3],[2,5],[4,9],[6,10],[7,11],[8,12]]
 => ?
 => ? => ? => ? = 22
[[1,4],[2,5],[3,7],[6,10],[8,11],[9,12]]
 => ?
 => ? => ? => ? = 18
[[1,4],[2,6],[3,9],[5,10],[7,11],[8,12]]
 => ?
 => ? => ? => ? = 20
[[1,4],[2,7],[3,8],[5,10],[6,11],[9,12]]
 => ?
 => ? => ? => ? = 18
[[1,5],[2,6],[3,7],[4,9],[8,11],[10,12]]
 => ?
 => ? => ? => ? = 14
[[1,5],[2,6],[3,8],[4,10],[7,11],[9,12]]
 => ?
 => ? => ? => ? = 16
[[1,5],[2,7],[3,9],[4,10],[6,11],[8,12]]
 => ?
 => ? => ? => ? = 18
[[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => 00000100000 => 00000100000 => ? = 6
[[1,2,3,9],[4,5,6],[7],[8]]
 => ?
 => ? => ? => ? = 25
[[1,2,5],[3,4,9],[6,7],[8]]
 => ?
 => ? => ? => ? = 23
[[1,3],[2,5],[4,9],[6],[7],[8]]
 => ?
 => ? => ? => ? = 13
[[1,2,3,7],[4,5,6,10],[8,9]]
 => ?
 => ? => ? => ? = 35
[]
 => []
 =>  =>  => ? = 0
[[1,3,4,5,6,7],[2,8,9,10]]
 => ?
 => ? => ? => ? = 33
[[1,3,4,5,6,7],[2,9,10],[8]]
 => ?
 => ? => ? => ? = 33
[[1,3,4,5,6],[2,8],[7],[9],[10]]
 => ?
 => ? => ? => ? = 29
[[1,3,4,5],[2,7],[6,9],[8],[10]]
 => ?
 => ? => ? => ? = 27
[[1,2,3,5,7,9,11,12],[4,6,8,10]]
 => [[1,4],[2,6],[3,8],[5,10],[7],[9],[11],[12]]
 => 11010101011 => 11010101011 => ? = 42
[[1,2,3,5,6,7,10,11,12],[4,8,9]]
 => [[1,4],[2,8],[3,9],[5],[6],[7],[10],[11],[12]]
 => 11011101111 => 11110111011 => ? = 52

Description

The sum of the positions of the ones in a binary word.

Matching statistic: St000009
(load all 3 compositions to match this statistic)

Mapping

St000009: Standard tableaux ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 98%

Values

[[1]]
 => 0
[[1,2]]
 => 1
[[1],[2]]
 => 0
[[1,2,3]]
 => 3
[[1,3],[2]]
 => 1
[[1,2],[3]]
 => 2
[[1],[2],[3]]
 => 0
[[1,2,3,4]]
 => 6
[[1,3,4],[2]]
 => 3
[[1,2,4],[3]]
 => 4
[[1,2,3],[4]]
 => 5
[[1,3],[2,4]]
 => 2
[[1,2],[3,4]]
 => 4
[[1,4],[2],[3]]
 => 1
[[1,3],[2],[4]]
 => 2
[[1,2],[3],[4]]
 => 3
[[1],[2],[3],[4]]
 => 0
[[1,2,3,4,5]]
 => 10
[[1,3,4,5],[2]]
 => 6
[[1,2,4,5],[3]]
 => 7
[[1,2,3,5],[4]]
 => 8
[[1,2,3,4],[5]]
 => 9
[[1,3,5],[2,4]]
 => 4
[[1,2,5],[3,4]]
 => 7
[[1,3,4],[2,5]]
 => 5
[[1,2,4],[3,5]]
 => 6
[[1,2,3],[4,5]]
 => 8
[[1,4,5],[2],[3]]
 => 3
[[1,3,5],[2],[4]]
 => 4
[[1,2,5],[3],[4]]
 => 5
[[1,3,4],[2],[5]]
 => 5
[[1,2,4],[3],[5]]
 => 6
[[1,2,3],[4],[5]]
 => 7
[[1,4],[2,5],[3]]
 => 2
[[1,3],[2,5],[4]]
 => 4
[[1,2],[3,5],[4]]
 => 5
[[1,3],[2,4],[5]]
 => 3
[[1,2],[3,4],[5]]
 => 6
[[1,5],[2],[3],[4]]
 => 1
[[1,4],[2],[3],[5]]
 => 2
[[1,3],[2],[4],[5]]
 => 3
[[1,2],[3],[4],[5]]
 => 4
[[1],[2],[3],[4],[5]]
 => 0
[[1,2,3,4,5,6]]
 => 15
[[1,3,4,5,6],[2]]
 => 10
[[1,2,4,5,6],[3]]
 => 11
[[1,2,3,5,6],[4]]
 => 12
[[1,2,3,4,6],[5]]
 => 13
[[1,2,3,4,5],[6]]
 => 14
[[1,3,5,6],[2,4]]
 => 7
[[1,2,3,4,5,9],[6],[7],[8]]
 => ? = 27
[[1,2,3,4,9],[5,6],[7],[8]]
 => ? = 26
[[1,2,3,9],[4,5,6],[7],[8]]
 => ? = 25
[[1,2,3,9],[4],[5],[6],[7],[8]]
 => ? = 16
[[1,2,5],[3,4,9],[6,7],[8]]
 => ? = 23
[[1,2,9],[3,4],[5],[6],[7],[8]]
 => ? = 15
[[1,3],[2,5],[4,9],[6],[7],[8]]
 => ? = 13
[[1,3],[2,9],[4],[5],[6],[7],[8]]
 => ? = 8
[[1,2,3,4,5,10],[6,7],[8,9]]
 => ? = 37
[[1,2,3,4,5,10],[6],[7],[8],[9]]
 => ? = 31
[[1,2,3,7],[4,5,6,10],[8,9]]
 => ? = 35
[[1,2,3,10],[4],[5],[6],[7],[8],[9]]
 => ? = 18
[[1,2,10],[3,4],[5],[6],[7],[8],[9]]
 => ? = 17
[[1,2,5,6],[3,7,8,9],[4]]
 => ? = 20
[[1,3,5,6,7,8,9,10],[2],[4]]
 => ? = 29
[[1,4,7,9],[2,5,8],[3,6,10]]
 => ? = 13
[[1,4,7,9],[2,8,10],[3],[5],[6]]
 => ? = 13
[[1,3,5],[2,6,8],[4,9],[7],[10]]
 => ? = 17
[[1,4,8],[2,6],[3,9],[5],[7],[10]]
 => ? = 15
[[1,3,4,5,6,7,8,9],[2,10]]
 => ? = 35
[[1,3,4,5,6,7,8,9],[2],[10]]
 => ? = 35
[[1,3,4,5,6,7],[2,8,9,10]]
 => ? = 33
[[1,3,4,5,6,7],[2,9,10],[8]]
 => ? = 33
[[1,3,4,5,6],[2,8],[7],[9],[10]]
 => ? = 29
[[1,3,4,5],[2,7],[6,9],[8,10]]
 => ? = 27
[[1,3,4,5],[2,7],[6,9],[8],[10]]
 => ? = 27
[[1,2,3,7,8,9],[4,5,6,10]]
 => ? = 37
[[1,2,3,7,8,9],[4,5,6],[10]]
 => ? = 37
[[1,4,7,8,9],[2,6],[3],[5],[10]]
 => ? = 21
[[1,3,8,9],[2,5],[4,7],[6,10]]
 => ? = 23
[[1,3,8,9],[2,5],[4,7],[6],[10]]
 => ? = 23
[[1,2,5,8,9],[3,7],[4],[6]]
 => ? = 19
[[1,2,5,6],[3,8,9],[4],[7]]
 => ? = 20
[[1,2,4],[3,6,7],[5,9],[8]]
 => ? = 22
[[1,2,7],[3,9],[4],[5],[6],[8]]
 => ? = 12
[[1,2],[3,6],[4,8],[5],[7],[9]]
 => ? = 14
[[1,2,3,8,9,10],[4,5],[6,7]]
 => ? = 33
[[1,2,8],[3,10],[4],[5],[6],[7],[9]]
 => ? = 13
[[1,4,7,9,11,12],[2,5,8,10],[3,6]]
 => ? = 22
[[1,5,9,11],[2,6,10,12],[3,7],[4,8]]
 => ? = 14
[[1,3],[2,5],[4,8],[6],[7],[9],[10]]
 => ? = 17
[[1,4],[2,7],[3,9],[5],[6],[8],[10]]
 => ? = 13
[[1,4],[2,6],[3,8],[5],[7],[9],[10]]
 => ? = 15
[[1,6],[2,7],[3],[4],[5],[8],[9],[10]]
 => ? = 5
[[1,6],[2],[3],[4],[5],[7],[8],[9],[10]]
 => ? = 5
[[1,3],[2,5],[4,7],[6,9],[8,11],[10],[12]]
 => ? = 30
[[1,3],[2,5],[4,9],[6,10],[7],[8],[11],[12]]
 => ? = 22
[[1,3],[2,5],[4,9],[6],[7],[8],[10],[11],[12]]
 => ? = 22
[[1,5],[2,7],[3,9],[4,10],[6],[8],[11],[12]]
 => ? = 18
[[1,5],[2,6],[3,9],[4,11],[7],[8],[10],[12]]
 => ? = 14

Description

The charge of a standard tableau.

Matching statistic: St000169

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 94%

Values

[[1]]
 => [[1]]
 => 0
[[1,2]]
 => [[1],[2]]
 => 1
[[1],[2]]
 => [[1,2]]
 => 0
[[1,2,3]]
 => [[1],[2],[3]]
 => 3
[[1,3],[2]]
 => [[1,2],[3]]
 => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => 2
[[1],[2],[3]]
 => [[1,2,3]]
 => 0
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 6
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 3
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => 4
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 5
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => 2
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 4
[[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => 2
[[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 3
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 10
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 6
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => 7
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => 8
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 9
[[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => 4
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 7
[[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => 5
[[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => 6
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 8
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 4
[[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => 5
[[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => 5
[[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 6
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 7
[[1,4],[2,5],[3]]
 => [[1,2,3],[4,5]]
 => 2
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 4
[[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => 5
[[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => 3
[[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 6
[[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => 2
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => 3
[[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 4
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 15
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 10
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => 11
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => 12
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => 13
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 14
[[1,3,5,6],[2,4]]
 => [[1,2],[3,4],[5],[6]]
 => 7
[[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
[[1,2,3,4,5],[6,7,8,9],[10,11,12]]
 => [[1,6,10],[2,7,11],[3,8,12],[4,9],[5]]
 => ? = 56
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
 => [[1,6,8,10,12],[2,7,9,11],[3],[4],[5]]
 => ? = 50
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
 => ? = 52
[[1,4,7,9],[2,5,8,10],[3,6]]
 => ?
 => ? = 13
[[1,3,5,8],[2,4,6,9],[7,10]]
 => ?
 => ? = 17
[[1,2],[3,6],[4,7],[5,8],[9,10]]
 => ?
 => ? = 15
[[1,3],[2,5],[4,8],[6,9],[7,10]]
 => ?
 => ? = 17
[[1,4],[2,5],[3,7],[6,9],[8,10]]
 => ?
 => ? = 13
[[1,3],[2,5],[4,7],[6,9],[8,11],[10,12]]
 => ?
 => ? = 30
[[1,3],[2,5],[4,9],[6,10],[7,11],[8,12]]
 => ?
 => ? = 22
[[1,4],[2,5],[3,7],[6,10],[8,11],[9,12]]
 => ?
 => ? = 18
[[1,4],[2,6],[3,9],[5,10],[7,11],[8,12]]
 => ?
 => ? = 20
[[1,4],[2,7],[3,8],[5,10],[6,11],[9,12]]
 => ?
 => ? = 18
[[1,5],[2,6],[3,7],[4,9],[8,11],[10,12]]
 => ?
 => ? = 14
[[1,5],[2,6],[3,8],[4,10],[7,11],[9,12]]
 => ?
 => ? = 16
[[1,5],[2,7],[3,9],[4,10],[6,11],[8,12]]
 => ?
 => ? = 18
[[1,2,3,4,5,9],[6],[7],[8]]
 => [[1,6,7,8],[2],[3],[4],[5],[9]]
 => ? = 27
[[1,2,3,4,9],[5,6],[7],[8]]
 => [[1,5,7,8],[2,6],[3],[4],[9]]
 => ? = 26
[[1,2,3,9],[4,5,6],[7],[8]]
 => ?
 => ? = 25
[[1,2,3,9],[4],[5],[6],[7],[8]]
 => [[1,4,5,6,7,8],[2],[3],[9]]
 => ? = 16
[[1,2,5],[3,4,9],[6,7],[8]]
 => ?
 => ? = 23
[[1,2,9],[3,4],[5],[6],[7],[8]]
 => [[1,3,5,6,7,8],[2,4],[9]]
 => ? = 15
[[1,2,9],[3],[4],[5],[6],[7],[8]]
 => [[1,3,4,5,6,7,8],[2],[9]]
 => ? = 9
[[1,3],[2,5],[4,9],[6],[7],[8]]
 => ?
 => ? = 13
[[1,3],[2,9],[4],[5],[6],[7],[8]]
 => [[1,2,4,5,6,7,8],[3,9]]
 => ? = 8
[[1,2,3,4,5,10],[6,7],[8,9]]
 => [[1,6,8],[2,7,9],[3],[4],[5],[10]]
 => ? = 37
[[1,2,3,4,5,10],[6],[7],[8],[9]]
 => [[1,6,7,8,9],[2],[3],[4],[5],[10]]
 => ? = 31
[[1,2,3,7],[4,5,6,10],[8,9]]
 => ?
 => ? = 35
[[1,2,3,10],[4],[5],[6],[7],[8],[9]]
 => [[1,4,5,6,7,8,9],[2],[3],[10]]
 => ? = 18
[[1,2,10],[3,4],[5],[6],[7],[8],[9]]
 => [[1,3,5,6,7,8,9],[2,4],[10]]
 => ? = 17
[[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
 => [[1,3,4,5,6,7,8,9],[2],[10]]
 => ? = 10
[[1,2,5,6],[3,7,8,9],[4]]
 => [[1,3,4],[2,7],[5,8],[6,9]]
 => ? = 20
[[1,3,5,6,7,8,9,10],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
 => ? = 29
[[1,4,7,9],[2,5,8],[3,6,10]]
 => [[1,2,3],[4,5,6],[7,8,10],[9]]
 => ? = 13
[[1,4,7,9],[2,8,10],[3],[5],[6]]
 => [[1,2,3,5,6],[4,8],[7,10],[9]]
 => ? = 13
[[1,3,5],[2,6,8],[4,9],[7],[10]]
 => [[1,2,4,7,10],[3,6,9],[5,8]]
 => ? = 17
[[1,4,8],[2,6],[3,9],[5],[7],[10]]
 => [[1,2,3,5,7,10],[4,6,9],[8]]
 => ? = 15
[[1,3,4,5,6,7,8,9],[2,10]]
 => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
 => ? = 35
[[1,3,4,5,6,7],[2,8,9,10]]
 => ?
 => ? = 33
[[1,3,4,5,6,7],[2,9,10],[8]]
 => ?
 => ? = 33
[[1,3,4,5,6],[2,8],[7],[9],[10]]
 => ?
 => ? = 29
[[1,3,4,5],[2,7],[6,9],[8,10]]
 => [[1,2,6,8],[3,7,9,10],[4],[5]]
 => ? = 27
[[1,3,4,5],[2,7],[6,9],[8],[10]]
 => ?
 => ? = 27
[[1,2,3,5,7,8,9,10],[4,6]]
 => [[1,4],[2,6],[3],[5],[7],[8],[9],[10]]
 => ? = 33
[[1,2,3,5,6,7,9,10],[4,8]]
 => [[1,4],[2,8],[3],[5],[6],[7],[9],[10]]
 => ? = 35
[[1,2,3,4,5,7,9,10],[6,8]]
 => [[1,6],[2,8],[3],[4],[5],[7],[9],[10]]
 => ? = 37
[[1,2,3,5,7,9,11,12],[4,6,8,10]]
 => [[1,4],[2,6],[3,8],[5,10],[7],[9],[11],[12]]
 => ? = 42
[[1,2,3,5,6,7,10,11,12],[4,8,9]]
 => [[1,4],[2,8],[3,9],[5],[6],[7],[10],[11],[12]]
 => ? = 52
[[1,2,3,5,6,7,8,9,11,12],[4,10]]
 => [[1,4],[2,10],[3],[5],[6],[7],[8],[9],[11],[12]]
 => ? = 54

Description

The cocharge of a standard tableau. The '''cocharge''' of a standard tableau

$T$ , denoted

$\mathrm{cc}(T)$ , is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation

$w_1 w_2\cdots w_n$ can be computed by the following algorithm: 1) Starting from

$w_n$ , scan the entries right-to-left until finding the entry

$1$ with a superscript

$0$ . 2) Continue scanning until the

$2$ is found, and label this with a superscript

$1$ . Then scan until the

$3$ is found, labeling with a

$2$ , and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.

Matching statistic: St000330

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 94%

Values

[[1]]
 => [[1]]
 => [[1]]
 => 0
[[1,2]]
 => [[1],[2]]
 => [[1],[2]]
 => 1
[[1],[2]]
 => [[1,2]]
 => [[1,2]]
 => 0
[[1,2,3]]
 => [[1],[2],[3]]
 => [[1],[2],[3]]
 => 3
[[1,3],[2]]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[[1],[2],[3]]
 => [[1,2,3]]
 => [[1,2,3]]
 => 0
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => [[1],[2],[3],[4]]
 => 6
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [[1,4],[2],[3]]
 => 3
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => [[1,3],[2],[4]]
 => 4
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => [[1,2],[3],[4]]
 => 5
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [[1,2],[3,4]]
 => 2
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [[1,3],[2,4]]
 => 4
[[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [[1,3,4],[2]]
 => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [[1,2,4],[3]]
 => 2
[[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => [[1,2,3],[4]]
 => 3
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [[1,2,3,4]]
 => 0
[[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => [[1],[2],[3],[4],[5]]
 => 10
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => [[1,5],[2],[3],[4]]
 => 6
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => [[1,4],[2],[3],[5]]
 => 7
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => [[1,3],[2],[4],[5]]
 => 8
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => [[1,2],[3],[4],[5]]
 => 9
[[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [[1,3],[2,5],[4]]
 => 4
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [[1,4],[2,5],[3]]
 => 7
[[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => [[1,2],[3,5],[4]]
 => 5
[[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => [[1,2],[3,4],[5]]
 => 6
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => [[1,3],[2,4],[5]]
 => 8
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => 4
[[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => [[1,3,4],[2],[5]]
 => 5
[[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => [[1,2,5],[3],[4]]
 => 5
[[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => 6
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 7
[[1,4],[2,5],[3]]
 => [[1,2,3],[4,5]]
 => [[1,2,5],[3,4]]
 => 2
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [[1,3,5],[2,4]]
 => 4
[[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => [[1,3,4],[2,5]]
 => 5
[[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => [[1,2,3],[4,5]]
 => 3
[[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => [[1,2,4],[3,5]]
 => 6
[[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => [[1,3,4,5],[2]]
 => 1
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => [[1,2,4,5],[3]]
 => 2
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => [[1,2,3,5],[4]]
 => 3
[[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => [[1,2,3,4],[5]]
 => 4
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => [[1,2,3,4,5]]
 => 0
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 15
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 10
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => [[1,5],[2],[3],[4],[6]]
 => 11
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => [[1,4],[2],[3],[5],[6]]
 => 12
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => [[1,3],[2],[4],[5],[6]]
 => 13
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2],[3],[4],[5],[6]]
 => 14
[[1,3,5,6],[2,4]]
 => [[1,2],[3,4],[5],[6]]
 => [[1,4],[2,6],[3],[5]]
 => 7
[[1,2,3,4,5,6,7,8,9,10,11,12]]
 => [[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
[[1,2,3,4,5],[6,7,8,9],[10,11,12]]
 => [[1,6,10],[2,7,11],[3,8,12],[4,9],[5]]
 => ?
 => ? = 56
[[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
 => [[1,6,8,10,12],[2,7,9,11],[3],[4],[5]]
 => ?
 => ? = 50
[[1,2,3,7,12],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
 => ?
 => ? = 52
[[1,4,7,9],[2,5,8,10],[3,6]]
 => ?
 => ?
 => ? = 13
[[1,3,5,8],[2,4,6,9],[7,10]]
 => ?
 => ?
 => ? = 17
[[1,2],[3,6],[4,7],[5,8],[9,10]]
 => ?
 => ?
 => ? = 15
[[1,3],[2,5],[4,8],[6,9],[7,10]]
 => ?
 => ?
 => ? = 17
[[1,4],[2,5],[3,7],[6,9],[8,10]]
 => ?
 => ?
 => ? = 13
[[1,3],[2,5],[4,7],[6,9],[8,11],[10,12]]
 => ?
 => ?
 => ? = 30
[[1,3],[2,5],[4,9],[6,10],[7,11],[8,12]]
 => ?
 => ?
 => ? = 22
[[1,4],[2,5],[3,7],[6,10],[8,11],[9,12]]
 => ?
 => ?
 => ? = 18
[[1,4],[2,6],[3,9],[5,10],[7,11],[8,12]]
 => ?
 => ?
 => ? = 20
[[1,4],[2,7],[3,8],[5,10],[6,11],[9,12]]
 => ?
 => ?
 => ? = 18
[[1,5],[2,6],[3,7],[4,9],[8,11],[10,12]]
 => ?
 => ?
 => ? = 14
[[1,5],[2,6],[3,8],[4,10],[7,11],[9,12]]
 => ?
 => ?
 => ? = 16
[[1,5],[2,7],[3,9],[4,10],[6,11],[8,12]]
 => ?
 => ?
 => ? = 18
[[1,2,3,4,5,9],[6],[7],[8]]
 => [[1,6,7,8],[2],[3],[4],[5],[9]]
 => [[1,3,4,5],[2],[6],[7],[8],[9]]
 => ? = 27
[[1,2,3,4,9],[5,6],[7],[8]]
 => [[1,5,7,8],[2,6],[3],[4],[9]]
 => [[1,3,4,6],[2,7],[5],[8],[9]]
 => ? = 26
[[1,2,3,9],[4,5,6],[7],[8]]
 => ?
 => ?
 => ? = 25
[[1,2,3,9],[4],[5],[6],[7],[8]]
 => [[1,4,5,6,7,8],[2],[3],[9]]
 => [[1,3,4,5,6,7],[2],[8],[9]]
 => ? = 16
[[1,2,5],[3,4,9],[6,7],[8]]
 => ?
 => ?
 => ? = 23
[[1,2,9],[3,4],[5],[6],[7],[8]]
 => [[1,3,5,6,7,8],[2,4],[9]]
 => [[1,3,4,5,6,8],[2,9],[7]]
 => ? = 15
[[1,2,9],[3],[4],[5],[6],[7],[8]]
 => [[1,3,4,5,6,7,8],[2],[9]]
 => [[1,3,4,5,6,7,8],[2],[9]]
 => ? = 9
[[1,3],[2,5],[4,9],[6],[7],[8]]
 => ?
 => ?
 => ? = 13
[[1,3],[2,9],[4],[5],[6],[7],[8]]
 => [[1,2,4,5,6,7,8],[3,9]]
 => [[1,3,4,5,6,7,9],[2,8]]
 => ? = 8
[[1,2,3,4,5,10],[6,7],[8,9]]
 => [[1,6,8],[2,7,9],[3],[4],[5],[10]]
 => ?
 => ? = 37
[[1,2,3,4,5,10],[6],[7],[8],[9]]
 => [[1,6,7,8,9],[2],[3],[4],[5],[10]]
 => [[1,3,4,5,6],[2],[7],[8],[9],[10]]
 => ? = 31
[[1,2,3,7],[4,5,6,10],[8,9]]
 => ?
 => ?
 => ? = 35
[[1,2,3,10],[4],[5],[6],[7],[8],[9]]
 => [[1,4,5,6,7,8,9],[2],[3],[10]]
 => [[1,3,4,5,6,7,8],[2],[9],[10]]
 => ? = 18
[[1,2,10],[3,4],[5],[6],[7],[8],[9]]
 => [[1,3,5,6,7,8,9],[2,4],[10]]
 => [[1,3,4,5,6,7,9],[2,10],[8]]
 => ? = 17
[[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
 => [[1,3,4,5,6,7,8,9],[2],[10]]
 => [[1,3,4,5,6,7,8,9],[2],[10]]
 => ? = 10
[[1,2,5,6],[3,7,8,9],[4]]
 => [[1,3,4],[2,7],[5,8],[6,9]]
 => [[1,4,8],[2,5],[3,7],[6,9]]
 => ? = 20
[[1,3,5,6,7,8,9,10],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
 => [[1,8,10],[2],[3],[4],[5],[6],[7],[9]]
 => ? = 29
[[1,4,7,9],[2,5,8],[3,6,10]]
 => [[1,2,3],[4,5,6],[7,8,10],[9]]
 => [[1,2,4],[3,6,7],[5,9,10],[8]]
 => ? = 13
[[1,4,7,9],[2,8,10],[3],[5],[6]]
 => [[1,2,3,5,6],[4,8],[7,10],[9]]
 => [[1,2,7,9,10],[3,4],[5,6],[8]]
 => ? = 13
[[1,3,5],[2,6,8],[4,9],[7],[10]]
 => [[1,2,4,7,10],[3,6,9],[5,8]]
 => [[1,2,3,6,10],[4,5,8],[7,9]]
 => ? = 17
[[1,4,8],[2,6],[3,9],[5],[7],[10]]
 => [[1,2,3,5,7,10],[4,6,9],[8]]
 => [[1,2,3,7,9,10],[4,5,8],[6]]
 => ? = 15
[[1,3,4,5,6,7,8,9],[2,10]]
 => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
 => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
 => ? = 35
[[1,3,4,5,6,7],[2,8,9,10]]
 => ?
 => ?
 => ? = 33
[[1,3,4,5,6,7],[2,9,10],[8]]
 => ?
 => ?
 => ? = 33
[[1,3,4,5,6],[2,8],[7],[9],[10]]
 => ?
 => ?
 => ? = 29
[[1,3,4,5],[2,7],[6,9],[8,10]]
 => [[1,2,6,8],[3,7,9,10],[4],[5]]
 => ?
 => ? = 27
[[1,3,4,5],[2,7],[6,9],[8],[10]]
 => ?
 => ?
 => ? = 27
[[1,2,3,5,7,8,9,10],[4,6]]
 => [[1,4],[2,6],[3],[5],[7],[8],[9],[10]]
 => [[1,6],[2,8],[3],[4],[5],[7],[9],[10]]
 => ? = 33
[[1,2,3,5,6,7,9,10],[4,8]]
 => [[1,4],[2,8],[3],[5],[6],[7],[9],[10]]
 => [[1,4],[2,8],[3],[5],[6],[7],[9],[10]]
 => ? = 35
[[1,2,3,4,5,7,9,10],[6,8]]
 => [[1,6],[2,8],[3],[4],[5],[7],[9],[10]]
 => [[1,4],[2,6],[3],[5],[7],[8],[9],[10]]
 => ? = 37
[[1,2,3,5,7,9,11,12],[4,6,8,10]]
 => [[1,4],[2,6],[3,8],[5,10],[7],[9],[11],[12]]
 => [[1,4],[2,6],[3,8],[5,10],[7],[9],[11],[12]]
 => ? = 42
[[1,2,3,5,6,7,10,11,12],[4,8,9]]
 => [[1,4],[2,8],[3,9],[5],[6],[7],[10],[11],[12]]
 => [[1,6],[2,7],[3,10],[4],[5],[8],[9],[11],[12]]
 => ? = 52
[[1,2,3,5,6,7,8,9,11,12],[4,10]]
 => [[1,4],[2,10],[3],[5],[6],[7],[8],[9],[11],[12]]
 => [[1,4],[2,10],[3],[5],[6],[7],[8],[9],[11],[12]]
 => ? = 54

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.

Matching statistic: St000446
(load all 12 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000446: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 31%

Values

[[1]]
 => [1] => [.,.]
 => [1] => 0
[[1,2]]
 => [1,2] => [.,[.,.]]
 => [2,1] => 1
[[1],[2]]
 => [2,1] => [[.,.],.]
 => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => 3
[[1,3],[2]]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 1
[[1,2],[3]]
 => [3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => 2
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 6
[[1,3,4],[2]]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 4
[[1,2,3],[4]]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 5
[[1,3],[2,4]]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 4
[[1,4],[2],[3]]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 3
[[1],[2],[3],[4]]
 => [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
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 6
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 7
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 8
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 9
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 4
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 7
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 5
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 6
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 8
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => 5
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => 6
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 7
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => 5
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => 3
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => 6
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => 2
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 4
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => 15
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => 10
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => [2,1,6,5,4,3] => 11
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
 => [3,2,1,6,5,4] => 12
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
 => [4,3,2,1,6,5] => 13
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
 => [5,4,3,2,1,6] => 14
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
 => [1,3,6,5,4,2] => 7
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ? = 21
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => ? = 15
[[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [2,1,7,6,5,4,3] => ? = 16
[[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 17
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 18
[[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [5,4,3,2,1,7,6] => ? = 19
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => [6,5,4,3,2,1,7] => ? = 20
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [2,1,7,6,5,4,3] => ? = 16
[[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 13
[[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 17
[[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ? = 13
[[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [2,1,5,4,7,6,3] => ? = 14
[[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [3,2,1,5,7,6,4] => ? = 15
[[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 18
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
 => [1,6,5,4,3,7,2] => ? = 14
[[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
 => [2,1,6,5,4,7,3] => ? = 15
[[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
 => [3,2,1,6,5,7,4] => ? = 16
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
 => [4,3,2,1,6,7,5] => ? = 17
[[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [5,4,3,2,1,7,6] => ? = 19
[[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [[[.,[.,.]],.],[.,[.,[.,.]]]]
 => [2,1,3,7,6,5,4] => ? = 12
[[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
 => [2,1,4,3,7,6,5] => ? = 13
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
 => [3,2,1,4,7,6,5] => ? = 14
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [[[.,.],[.,[.,[.,.]]]],[.,.]]
 => [1,5,4,3,2,7,6] => ? = 13
[[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => [2,1,5,4,3,7,6] => ? = 14
[[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [[[.,[.,[.,.]]],[.,.]],[.,.]]
 => [3,2,1,5,4,7,6] => ? = 15
[[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [4,3,2,1,5,7,6] => ? = 16
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
 => [1,6,5,4,3,2,7] => ? = 14
[[1,2,4,5,6],[3],[7]]
 => [7,3,1,2,4,5,6] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
 => [2,1,6,5,4,3,7] => ? = 15
[[1,2,3,5,6],[4],[7]]
 => [7,4,1,2,3,5,6] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
 => [3,2,1,6,5,4,7] => ? = 16
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
 => [4,3,2,1,6,5,7] => ? = 17
[[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [[[.,[.,[.,[.,[.,.]]]]],.],.]
 => [5,4,3,2,1,6,7] => ? = 18
[[1,2,5,7],[3,4,6]]
 => [3,4,6,1,2,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
 => [2,1,5,7,6,4,3] => ? = 14
[[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
 => [2,1,4,7,6,5,3] => ? = 13
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
 => [3,2,1,7,6,5,4] => ? = 17
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
 => [2,1,6,5,7,4,3] => ? = 15
[[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
 => [2,1,4,6,7,5,3] => ? = 12
[[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
 => [3,2,1,6,7,5,4] => ? = 16
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
 => [1,5,4,3,7,6,2] => ? = 13
[[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
 => [2,1,5,4,7,6,3] => ? = 14
[[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [3,2,1,5,7,6,4] => ? = 15
[[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
 => [4,3,2,1,7,6,5] => ? = 18
[[1,2,6,7],[3,5],[4]]
 => [4,3,5,1,2,6,7] => [[[.,[.,.]],.],[.,[.,[.,.]]]]
 => [2,1,3,7,6,5,4] => ? = 12
[[1,2,6,7],[3,4],[5]]
 => [5,3,4,1,2,6,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
 => [2,1,4,3,7,6,5] => ? = 13
[[1,2,5,7],[3,6],[4]]
 => [4,3,6,1,2,5,7] => [[[.,[.,.]],.],[[.,.],[.,.]]]
 => [2,1,3,5,7,6,4] => ? = 10
[[1,2,4,7],[3,6],[5]]
 => [5,3,6,1,2,4,7] => [[[.,[.,.]],[.,.]],[.,[.,.]]]
 => [2,1,4,3,7,6,5] => ? = 13
[[1,2,3,7],[4,6],[5]]
 => [5,4,6,1,2,3,7] => [[[.,[.,[.,.]]],.],[.,[.,.]]]
 => [3,2,1,4,7,6,5] => ? = 14
[[1,2,5,7],[3,4],[6]]
 => [6,3,4,1,2,5,7] => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => [2,1,5,4,3,7,6] => ? = 14
[[1,2,4,7],[3,5],[6]]
 => [6,3,5,1,2,4,7] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => [2,1,4,5,3,7,6] => ? = 11
[[1,2,3,7],[4,5],[6]]
 => [6,4,5,1,2,3,7] => [[[.,[.,[.,.]]],[.,.]],[.,.]]
 => [3,2,1,5,4,7,6] => ? = 15
[[1,2,5,6],[3,7],[4]]
 => [4,3,7,1,2,5,6] => [[[.,[.,.]],.],[[.,[.,.]],.]]
 => [2,1,3,6,5,7,4] => ? = 11

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: St000004
(load all 2 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 31%

Values

[[1]]
 => [1] => [1] => [1] => 0
[[1,2]]
 => [1,2] => [2,1] => [2,1] => 1
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [3,2,1] => [3,2,1] => 3
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => [3,1,2] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => [2,3,1] => 2
[[1],[2],[3]]
 => [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
[[1,3,4],[2]]
 => [2,1,3,4] => [3,4,2,1] => [4,3,1,2] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[[1,2,3],[4]]
 => [4,1,2,3] => [1,4,3,2] => [3,4,2,1] => 5
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => [3,2,4,1] => 4
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,4,1] => [4,1,2,3] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,4,2] => [2,4,1,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,4,3] => [2,3,4,1] => 3
[[1],[2],[3],[4]]
 => [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
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [4,5,3,2,1] => [5,4,3,1,2] => 6
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,5,4,2,1] => [5,4,2,3,1] => 7
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,5,4,3,1] => [5,3,4,2,1] => 8
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,5,4,3,2] => [4,5,3,2,1] => 9
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [4,2,5,3,1] => [5,3,4,1,2] => 4
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,2,5,4,1] => [5,3,2,4,1] => 7
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [4,1,5,3,2] => [4,5,3,1,2] => 5
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,1,5,4,2] => [4,5,2,3,1] => 6
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,5,4,3] => [4,3,5,2,1] => 8
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,4,5,2,1] => [5,4,1,2,3] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,5,3,1] => [5,2,4,1,3] => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [2,3,5,4,1] => [5,2,3,4,1] => 5
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,4,5,3,2] => [3,5,4,1,2] => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,4,2] => [3,5,2,4,1] => 6
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => [3,4,5,2,1] => 7
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,4,1,5,2] => [4,5,1,2,3] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => [4,2,5,1,3] => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [2,3,1,5,4] => [4,2,3,5,1] => 5
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,4,2,5,3] => [3,4,5,1,2] => 3
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,2,5,4] => [3,4,2,5,1] => 6
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [2,3,4,5,1] => [5,1,2,3,4] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,3,4,5,2] => [2,5,1,3,4] => 2
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,4,5,3] => [2,3,5,1,4] => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,5,4] => [2,3,4,5,1] => 4
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 15
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 10
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [4,6,5,3,2,1] => [6,5,4,2,3,1] => 11
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [3,6,5,4,2,1] => [6,5,3,4,2,1] => 12
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,6,5,4,3,1] => [6,4,5,3,2,1] => 13
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => [5,6,4,3,2,1] => 14
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [5,3,6,4,2,1] => [6,5,3,4,1,2] => 7
[[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
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 15
[[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => [7,6,5,4,2,3,1] => ? = 16
[[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => [7,6,5,3,4,2,1] => ? = 17
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => [7,6,4,5,3,2,1] => ? = 18
[[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => [7,5,6,4,3,2,1] => ? = 19
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [6,7,5,4,3,2,1] => ? = 20
[[1,3,5,6,7],[2,4]]
 => [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => [7,6,5,3,4,1,2] => ? = 11
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => [7,6,5,3,2,4,1] => ? = 16
[[1,3,4,6,7],[2,5]]
 => [2,5,1,3,4,6,7] => [6,3,7,5,4,2,1] => [7,6,4,5,3,1,2] => ? = 12
[[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [5,3,7,6,4,2,1] => [7,6,4,5,2,3,1] => ? = 13
[[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [4,3,7,6,5,2,1] => [7,6,4,3,5,2,1] => ? = 17
[[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => [6,2,7,5,4,3,1] => [7,5,6,4,3,1,2] => ? = 13
[[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => [5,2,7,6,4,3,1] => [7,5,6,4,2,3,1] => ? = 14
[[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => [4,2,7,6,5,3,1] => [7,5,6,3,4,2,1] => ? = 15
[[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [3,2,7,6,5,4,1] => [7,5,4,6,3,2,1] => ? = 18
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [6,1,7,5,4,3,2] => [6,7,5,4,3,1,2] => ? = 14
[[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [5,1,7,6,4,3,2] => [6,7,5,4,2,3,1] => ? = 15
[[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [4,1,7,6,5,3,2] => [6,7,5,3,4,2,1] => ? = 16
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [3,1,7,6,5,4,2] => [6,7,4,5,3,2,1] => ? = 17
[[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => [6,5,7,4,3,2,1] => ? = 19
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 10
[[1,3,5,6,7],[2],[4]]
 => [4,2,1,3,5,6,7] => [4,6,7,5,3,2,1] => [7,6,5,2,4,1,3] => ? = 11
[[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,5,7,6,3,2,1] => [7,6,5,2,3,4,1] => ? = 12
[[1,3,4,6,7],[2],[5]]
 => [5,2,1,3,4,6,7] => [3,6,7,5,4,2,1] => [7,6,3,5,4,1,2] => ? = 12
[[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [3,5,7,6,4,2,1] => [7,6,3,5,2,4,1] => ? = 13
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [3,4,7,6,5,2,1] => [7,6,3,4,5,2,1] => ? = 14
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [2,6,7,5,4,3,1] => [7,4,6,5,3,1,2] => ? = 13
[[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [2,5,7,6,4,3,1] => [7,4,6,5,2,3,1] => ? = 14
[[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [2,4,7,6,5,3,1] => [7,4,6,3,5,2,1] => ? = 15
[[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [2,3,7,6,5,4,1] => [7,4,5,6,3,2,1] => ? = 16
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [1,6,7,5,4,3,2] => [5,7,6,4,3,1,2] => ? = 14
[[1,2,4,5,6],[3],[7]]
 => [7,3,1,2,4,5,6] => [1,5,7,6,4,3,2] => [5,7,6,4,2,3,1] => ? = 15
[[1,2,3,5,6],[4],[7]]
 => [7,4,1,2,3,5,6] => [1,4,7,6,5,3,2] => [5,7,6,3,4,2,1] => ? = 16
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [1,3,7,6,5,4,2] => [5,7,4,6,3,2,1] => ? = 17
[[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => [5,6,7,4,3,2,1] => ? = 18
[[1,3,5,7],[2,4,6]]
 => [2,4,6,1,3,5,7] => [6,4,2,7,5,3,1] => [7,5,6,3,4,1,2] => ? = 9
[[1,2,5,7],[3,4,6]]
 => [3,4,6,1,2,5,7] => [5,4,2,7,6,3,1] => [7,5,6,3,2,4,1] => ? = 14
[[1,3,4,7],[2,5,6]]
 => [2,5,6,1,3,4,7] => [6,3,2,7,5,4,1] => [7,5,4,6,3,1,2] => ? = 12
[[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => [5,3,2,7,6,4,1] => [7,5,4,6,2,3,1] => ? = 13
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => [7,5,4,3,6,2,1] => ? = 17
[[1,3,5,6],[2,4,7]]
 => [2,4,7,1,3,5,6] => [6,4,1,7,5,3,2] => [6,7,5,3,4,1,2] => ? = 10
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [5,4,1,7,6,3,2] => [6,7,5,3,2,4,1] => ? = 15
[[1,3,4,6],[2,5,7]]
 => [2,5,7,1,3,4,6] => [6,3,1,7,5,4,2] => [6,7,4,5,3,1,2] => ? = 11
[[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => [5,3,1,7,6,4,2] => [6,7,4,5,2,3,1] => ? = 12
[[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => [4,3,1,7,6,5,2] => [6,7,4,3,5,2,1] => ? = 16
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => [6,2,1,7,5,4,3] => [6,5,7,4,3,1,2] => ? = 13
[[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [5,2,1,7,6,4,3] => [6,5,7,4,2,3,1] => ? = 14
[[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [4,2,1,7,6,5,3] => [6,5,7,3,4,2,1] => ? = 15
[[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => [6,5,4,7,3,2,1] => ? = 18

Description

The major index of a permutation. This is the sum of the positions of its descents,

$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$ Its generating function is

$[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for

$[k]_q = 1 + q + q^2 + \dots q^{k-1}$ . A statistic equidistributed with the major index is called '''Mahonian statistic'''.

Matching statistic: St000304
(load all 11 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000304: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 31%

Values

[[1]]
 => [1] => 0
[[1,2]]
 => [1,2] => 1
[[1],[2]]
 => [2,1] => 0
[[1,2,3]]
 => [1,2,3] => 3
[[1,3],[2]]
 => [2,1,3] => 1
[[1,2],[3]]
 => [3,1,2] => 2
[[1],[2],[3]]
 => [3,2,1] => 0
[[1,2,3,4]]
 => [1,2,3,4] => 6
[[1,3,4],[2]]
 => [2,1,3,4] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => 4
[[1,2,3],[4]]
 => [4,1,2,3] => 5
[[1,3],[2,4]]
 => [2,4,1,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => 4
[[1,4],[2],[3]]
 => [3,2,1,4] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 10
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => 6
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => 7
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => 8
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => 9
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => 4
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => 7
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => 5
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => 6
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => 8
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 5
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 6
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 7
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 5
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 3
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 6
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => 2
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 15
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 10
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => 11
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => 12
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => 13
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 14
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => 7
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ? = 21
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 15
[[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => ? = 16
[[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => ? = 17
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => ? = 18
[[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => ? = 19
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 20
[[1,3,5,6,7],[2,4]]
 => [2,4,1,3,5,6,7] => ? = 11
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 16
[[1,3,4,6,7],[2,5]]
 => [2,5,1,3,4,6,7] => ? = 12
[[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => ? = 13
[[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => ? = 17
[[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => ? = 13
[[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => ? = 14
[[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => ? = 15
[[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => ? = 18
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => ? = 14
[[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => ? = 15
[[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => ? = 16
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => ? = 17
[[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 19
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 10
[[1,3,5,6,7],[2],[4]]
 => [4,2,1,3,5,6,7] => ? = 11
[[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => ? = 12
[[1,3,4,6,7],[2],[5]]
 => [5,2,1,3,4,6,7] => ? = 12
[[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => ? = 13
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => ? = 14
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => ? = 13
[[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => ? = 14
[[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => ? = 15
[[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => ? = 16
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => ? = 14
[[1,2,4,5,6],[3],[7]]
 => [7,3,1,2,4,5,6] => ? = 15
[[1,2,3,5,6],[4],[7]]
 => [7,4,1,2,3,5,6] => ? = 16
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => ? = 17
[[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ? = 18
[[1,3,5,7],[2,4,6]]
 => [2,4,6,1,3,5,7] => ? = 9
[[1,2,5,7],[3,4,6]]
 => [3,4,6,1,2,5,7] => ? = 14
[[1,3,4,7],[2,5,6]]
 => [2,5,6,1,3,4,7] => ? = 12
[[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => ? = 13
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 17
[[1,3,5,6],[2,4,7]]
 => [2,4,7,1,3,5,6] => ? = 10
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => ? = 15
[[1,3,4,6],[2,5,7]]
 => [2,5,7,1,3,4,6] => ? = 11
[[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => ? = 12
[[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => ? = 16
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => ? = 13
[[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => ? = 14
[[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => ? = 15
[[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ? = 18

Description

The load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.

Matching statistic: St000305
(load all 5 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 31%

Values

[[1]]
 => [1] => [1] => 0
[[1,2]]
 => [1,2] => [2,1] => 1
[[1],[2]]
 => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [3,2,1] => 3
[[1,3],[2]]
 => [2,1,3] => [2,3,1] => 1
[[1,2],[3]]
 => [3,1,2] => [1,3,2] => 2
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 6
[[1,3,4],[2]]
 => [2,1,3,4] => [3,4,2,1] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => [2,4,3,1] => 4
[[1,2,3],[4]]
 => [4,1,2,3] => [1,4,3,2] => 5
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 4
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,4,1] => 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,4,2] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,4,3] => 3
[[1],[2],[3],[4]]
 => [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
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [4,5,3,2,1] => 6
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [3,5,4,2,1] => 7
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,5,4,3,1] => 8
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,5,4,3,2] => 9
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [4,2,5,3,1] => 4
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,2,5,4,1] => 7
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [4,1,5,3,2] => 5
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,1,5,4,2] => 6
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,5,4,3] => 8
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [2,4,5,3,1] => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [2,3,5,4,1] => 5
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,4,5,3,2] => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,3,5,4,2] => 6
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 7
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [3,4,1,5,2] => 2
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [2,3,1,5,4] => 5
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,4,2,5,3] => 3
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,2,5,4] => 6
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [2,3,4,5,1] => 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,3,4,5,2] => 2
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,2,4,5,3] => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,5,4] => 4
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 15
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [5,6,4,3,2,1] => 10
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [4,6,5,3,2,1] => 11
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [3,6,5,4,2,1] => 12
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,6,5,4,3,1] => 13
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => 14
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [5,3,6,4,2,1] => 7
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 21
[[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 15
[[1,2,4,5,6,7],[3]]
 => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? = 16
[[1,2,3,5,6,7],[4]]
 => [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => ? = 17
[[1,2,3,4,6,7],[5]]
 => [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => ? = 18
[[1,2,3,4,5,7],[6]]
 => [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => ? = 19
[[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 20
[[1,3,5,6,7],[2,4]]
 => [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => ? = 11
[[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => ? = 16
[[1,3,4,6,7],[2,5]]
 => [2,5,1,3,4,6,7] => [6,3,7,5,4,2,1] => ? = 12
[[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [5,3,7,6,4,2,1] => ? = 13
[[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [4,3,7,6,5,2,1] => ? = 17
[[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => [6,2,7,5,4,3,1] => ? = 13
[[1,2,4,5,7],[3,6]]
 => [3,6,1,2,4,5,7] => [5,2,7,6,4,3,1] => ? = 14
[[1,2,3,5,7],[4,6]]
 => [4,6,1,2,3,5,7] => [4,2,7,6,5,3,1] => ? = 15
[[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [3,2,7,6,5,4,1] => ? = 18
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [6,1,7,5,4,3,2] => ? = 14
[[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [5,1,7,6,4,3,2] => ? = 15
[[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [4,1,7,6,5,3,2] => ? = 16
[[1,2,3,4,6],[5,7]]
 => [5,7,1,2,3,4,6] => [3,1,7,6,5,4,2] => ? = 17
[[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => ? = 19
[[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => ? = 10
[[1,3,5,6,7],[2],[4]]
 => [4,2,1,3,5,6,7] => [4,6,7,5,3,2,1] => ? = 11
[[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [4,5,7,6,3,2,1] => ? = 12
[[1,3,4,6,7],[2],[5]]
 => [5,2,1,3,4,6,7] => [3,6,7,5,4,2,1] => ? = 12
[[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [3,5,7,6,4,2,1] => ? = 13
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [3,4,7,6,5,2,1] => ? = 14
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [2,6,7,5,4,3,1] => ? = 13
[[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [2,5,7,6,4,3,1] => ? = 14
[[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [2,4,7,6,5,3,1] => ? = 15
[[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [2,3,7,6,5,4,1] => ? = 16
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [1,6,7,5,4,3,2] => ? = 14
[[1,2,4,5,6],[3],[7]]
 => [7,3,1,2,4,5,6] => [1,5,7,6,4,3,2] => ? = 15
[[1,2,3,5,6],[4],[7]]
 => [7,4,1,2,3,5,6] => [1,4,7,6,5,3,2] => ? = 16
[[1,2,3,4,6],[5],[7]]
 => [7,5,1,2,3,4,6] => [1,3,7,6,5,4,2] => ? = 17
[[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => ? = 18
[[1,3,5,7],[2,4,6]]
 => [2,4,6,1,3,5,7] => [6,4,2,7,5,3,1] => ? = 9
[[1,2,5,7],[3,4,6]]
 => [3,4,6,1,2,5,7] => [5,4,2,7,6,3,1] => ? = 14
[[1,3,4,7],[2,5,6]]
 => [2,5,6,1,3,4,7] => [6,3,2,7,5,4,1] => ? = 12
[[1,2,4,7],[3,5,6]]
 => [3,5,6,1,2,4,7] => [5,3,2,7,6,4,1] => ? = 13
[[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => ? = 17
[[1,3,5,6],[2,4,7]]
 => [2,4,7,1,3,5,6] => [6,4,1,7,5,3,2] => ? = 10
[[1,2,5,6],[3,4,7]]
 => [3,4,7,1,2,5,6] => [5,4,1,7,6,3,2] => ? = 15
[[1,3,4,6],[2,5,7]]
 => [2,5,7,1,3,4,6] => [6,3,1,7,5,4,2] => ? = 11
[[1,2,4,6],[3,5,7]]
 => [3,5,7,1,2,4,6] => [5,3,1,7,6,4,2] => ? = 12
[[1,2,3,6],[4,5,7]]
 => [4,5,7,1,2,3,6] => [4,3,1,7,6,5,2] => ? = 16
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => [6,2,1,7,5,4,3] => ? = 13
[[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [5,2,1,7,6,4,3] => ? = 14
[[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [4,2,1,7,6,5,3] => ? = 15
[[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 18

Description

The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].

Matching statistic: St000833
(load all 2 compositions to match this statistic)

Mapping

Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000833: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 31%

Values

[[1]]
 => [[1]]
 => [1] => [1] => ? = 0
[[1,2]]
 => [[1],[2]]
 => [2,1] => [2,1] => 1
[[1],[2]]
 => [[1,2]]
 => [1,2] => [1,2] => 0
[[1,2,3]]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 3
[[1,3],[2]]
 => [[1,2],[3]]
 => [3,1,2] => [2,3,1] => 1
[[1,2],[3]]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
[[1],[2],[3]]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 6
[[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => 3
[[1,2,4],[3]]
 => [[1,3],[2],[4]]
 => [4,2,1,3] => [3,2,4,1] => 4
[[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 5
[[1,3],[2,4]]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 2
[[1,2],[3,4]]
 => [[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => 4
[[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [2,3,4,1] => 1
[[1,3],[2],[4]]
 => [[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => 2
[[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 3
[[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => [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
[[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [4,5,3,2,1] => 6
[[1,2,4,5],[3]]
 => [[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,3,5,2,1] => 7
[[1,2,3,5],[4]]
 => [[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,2,5,1] => 8
[[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => 9
[[1,3,5],[2,4]]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [4,5,2,3,1] => 4
[[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [4,2,5,3,1] => 7
[[1,3,4],[2,5]]
 => [[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,2,1,3] => 5
[[1,2,4],[3,5]]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => 6
[[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,2,1,5,3] => 8
[[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
 => [[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,4,2,5,1] => 4
[[1,2,5],[3],[4]]
 => [[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,2,4,5,1] => 5
[[1,3,4],[2],[5]]
 => [[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,4,2,1,5] => 5
[[1,2,4],[3],[5]]
 => [[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,2,4,1,5] => 6
[[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => 7
[[1,4],[2,5],[3]]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,4,5,1,2] => 2
[[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,4,1,5,2] => 4
[[1,2],[3,5],[4]]
 => [[1,3,4],[2,5]]
 => [2,5,1,3,4] => [3,1,4,5,2] => 5
[[1,3],[2,4],[5]]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => 3
[[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => [2,4,1,3,5] => [3,1,4,2,5] => 6
[[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,3,4,5,1] => 1
[[1,4],[2],[3],[5]]
 => [[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => 2
[[1,3],[2],[4],[5]]
 => [[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => 3
[[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 4
[[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 15
[[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 10
[[1,2,4,5,6],[3]]
 => [[1,3],[2],[4],[5],[6]]
 => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 11
[[1,2,3,5,6],[4]]
 => [[1,4],[2],[3],[5],[6]]
 => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 12
[[1,2,3,4,6],[5]]
 => [[1,5],[2],[3],[4],[6]]
 => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 13
[[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 14
[[1,3,5,6],[2,4]]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [5,6,3,4,2,1] => 7
[[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => [6,5,2,4,1,3] => [5,3,6,4,2,1] => 11
[[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
[[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 15
[[1,2,4,5,6,7],[3]]
 => [[1,3],[2],[4],[5],[6],[7]]
 => [7,6,5,4,2,1,3] => [6,5,7,4,3,2,1] => ? = 16
[[1,2,3,5,6,7],[4]]
 => [[1,4],[2],[3],[5],[6],[7]]
 => [7,6,5,3,2,1,4] => [6,5,4,7,3,2,1] => ? = 17
[[1,2,3,4,6,7],[5]]
 => [[1,5],[2],[3],[4],[6],[7]]
 => [7,6,4,3,2,1,5] => [6,5,4,3,7,2,1] => ? = 18
[[1,2,3,4,5,7],[6]]
 => [[1,6],[2],[3],[4],[5],[7]]
 => [7,5,4,3,2,1,6] => [6,5,4,3,2,7,1] => ? = 19
[[1,2,3,4,5,6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 20
[[1,3,5,6,7],[2,4]]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [6,7,4,5,3,2,1] => ? = 11
[[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => [7,6,5,2,4,1,3] => [6,4,7,5,3,2,1] => ? = 16
[[1,3,4,6,7],[2,5]]
 => [[1,2],[3,5],[4],[6],[7]]
 => [7,6,4,3,5,1,2] => [6,7,4,3,5,2,1] => ? = 12
[[1,2,4,6,7],[3,5]]
 => [[1,3],[2,5],[4],[6],[7]]
 => [7,6,4,2,5,1,3] => [6,4,7,3,5,2,1] => ? = 13
[[1,2,3,6,7],[4,5]]
 => [[1,4],[2,5],[3],[6],[7]]
 => [7,6,3,2,5,1,4] => [6,4,3,7,5,2,1] => ? = 17
[[1,3,4,5,7],[2,6]]
 => [[1,2],[3,6],[4],[5],[7]]
 => [7,5,4,3,6,1,2] => [6,7,4,3,2,5,1] => ? = 13
[[1,2,4,5,7],[3,6]]
 => [[1,3],[2,6],[4],[5],[7]]
 => [7,5,4,2,6,1,3] => [6,4,7,3,2,5,1] => ? = 14
[[1,2,3,5,7],[4,6]]
 => [[1,4],[2,6],[3],[5],[7]]
 => [7,5,3,2,6,1,4] => [6,4,3,7,2,5,1] => ? = 15
[[1,2,3,4,7],[5,6]]
 => [[1,5],[2,6],[3],[4],[7]]
 => [7,4,3,2,6,1,5] => [6,4,3,2,7,5,1] => ? = 18
[[1,3,4,5,6],[2,7]]
 => [[1,2],[3,7],[4],[5],[6]]
 => [6,5,4,3,7,1,2] => [6,7,4,3,2,1,5] => ? = 14
[[1,2,4,5,6],[3,7]]
 => [[1,3],[2,7],[4],[5],[6]]
 => [6,5,4,2,7,1,3] => [6,4,7,3,2,1,5] => ? = 15
[[1,2,3,5,6],[4,7]]
 => [[1,4],[2,7],[3],[5],[6]]
 => [6,5,3,2,7,1,4] => [6,4,3,7,2,1,5] => ? = 16
[[1,2,3,4,6],[5,7]]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [6,4,3,2,7,1,5] => ? = 17
[[1,2,3,4,5],[6,7]]
 => [[1,6],[2,7],[3],[4],[5]]
 => [5,4,3,2,7,1,6] => [6,4,3,2,1,7,5] => ? = 19
[[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 10
[[1,3,5,6,7],[2],[4]]
 => [[1,2,4],[3],[5],[6],[7]]
 => [7,6,5,3,1,2,4] => [5,6,4,7,3,2,1] => ? = 11
[[1,2,5,6,7],[3],[4]]
 => [[1,3,4],[2],[5],[6],[7]]
 => [7,6,5,2,1,3,4] => [5,4,6,7,3,2,1] => ? = 12
[[1,3,4,6,7],[2],[5]]
 => [[1,2,5],[3],[4],[6],[7]]
 => [7,6,4,3,1,2,5] => [5,6,4,3,7,2,1] => ? = 12
[[1,2,4,6,7],[3],[5]]
 => [[1,3,5],[2],[4],[6],[7]]
 => [7,6,4,2,1,3,5] => [5,4,6,3,7,2,1] => ? = 13
[[1,2,3,6,7],[4],[5]]
 => [[1,4,5],[2],[3],[6],[7]]
 => [7,6,3,2,1,4,5] => [5,4,3,6,7,2,1] => ? = 14
[[1,3,4,5,7],[2],[6]]
 => [[1,2,6],[3],[4],[5],[7]]
 => [7,5,4,3,1,2,6] => [5,6,4,3,2,7,1] => ? = 13
[[1,2,4,5,7],[3],[6]]
 => [[1,3,6],[2],[4],[5],[7]]
 => [7,5,4,2,1,3,6] => [5,4,6,3,2,7,1] => ? = 14
[[1,2,3,5,7],[4],[6]]
 => [[1,4,6],[2],[3],[5],[7]]
 => [7,5,3,2,1,4,6] => [5,4,3,6,2,7,1] => ? = 15
[[1,2,3,4,7],[5],[6]]
 => [[1,5,6],[2],[3],[4],[7]]
 => [7,4,3,2,1,5,6] => [5,4,3,2,6,7,1] => ? = 16
[[1,3,4,5,6],[2],[7]]
 => [[1,2,7],[3],[4],[5],[6]]
 => [6,5,4,3,1,2,7] => [5,6,4,3,2,1,7] => ? = 14
[[1,2,4,5,6],[3],[7]]
 => [[1,3,7],[2],[4],[5],[6]]
 => [6,5,4,2,1,3,7] => [5,4,6,3,2,1,7] => ? = 15
[[1,2,3,5,6],[4],[7]]
 => [[1,4,7],[2],[3],[5],[6]]
 => [6,5,3,2,1,4,7] => [5,4,3,6,2,1,7] => ? = 16
[[1,2,3,4,6],[5],[7]]
 => [[1,5,7],[2],[3],[4],[6]]
 => [6,4,3,2,1,5,7] => [5,4,3,2,6,1,7] => ? = 17
[[1,2,3,4,5],[6],[7]]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 18
[[1,3,5,7],[2,4,6]]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [6,7,4,5,2,3,1] => ? = 9
[[1,2,5,7],[3,4,6]]
 => [[1,3],[2,4],[5,6],[7]]
 => [7,5,6,2,4,1,3] => [6,4,7,5,2,3,1] => ? = 14
[[1,3,4,7],[2,5,6]]
 => [[1,2],[3,5],[4,6],[7]]
 => [7,4,6,3,5,1,2] => [6,7,4,2,5,3,1] => ? = 12
[[1,2,4,7],[3,5,6]]
 => [[1,3],[2,5],[4,6],[7]]
 => [7,4,6,2,5,1,3] => [6,4,7,2,5,3,1] => ? = 13
[[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => [7,3,6,2,5,1,4] => [6,4,2,7,5,3,1] => ? = 17
[[1,3,5,6],[2,4,7]]
 => [[1,2],[3,4],[5,7],[6]]
 => [6,5,7,3,4,1,2] => [6,7,4,5,2,1,3] => ? = 10
[[1,2,5,6],[3,4,7]]
 => [[1,3],[2,4],[5,7],[6]]
 => [6,5,7,2,4,1,3] => [6,4,7,5,2,1,3] => ? = 15
[[1,3,4,6],[2,5,7]]
 => [[1,2],[3,5],[4,7],[6]]
 => [6,4,7,3,5,1,2] => [6,7,4,2,5,1,3] => ? = 11
[[1,2,4,6],[3,5,7]]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => ? = 12
[[1,2,3,6],[4,5,7]]
 => [[1,4],[2,5],[3,7],[6]]
 => [6,3,7,2,5,1,4] => [6,4,2,7,5,1,3] => ? = 16
[[1,3,4,5],[2,6,7]]
 => [[1,2],[3,6],[4,7],[5]]
 => [5,4,7,3,6,1,2] => [6,7,4,2,1,5,3] => ? = 13
[[1,2,4,5],[3,6,7]]
 => [[1,3],[2,6],[4,7],[5]]
 => [5,4,7,2,6,1,3] => [6,4,7,2,1,5,3] => ? = 14
[[1,2,3,5],[4,6,7]]
 => [[1,4],[2,6],[3,7],[5]]
 => [5,3,7,2,6,1,4] => [6,4,2,7,1,5,3] => ? = 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

$n$ .

The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000102The charge of a semistandard tableau. St000101The cocharge of a semistandard tableau.