Processing math: 0%

Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000009
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => 0
[2]
 => [[1,2]]
 => 1
[1,1]
 => [[1],[2]]
 => 0
[3]
 => [[1,2,3]]
 => 3
[2,1]
 => [[1,2],[3]]
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => 0
[4]
 => [[1,2,3,4]]
 => 6
[3,1]
 => [[1,2,3],[4]]
 => 5
[2,2]
 => [[1,2],[3,4]]
 => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => 0
[5]
 => [[1,2,3,4,5]]
 => 10
[4,1]
 => [[1,2,3,4],[5]]
 => 9
[3,2]
 => [[1,2,3],[4,5]]
 => 8
[3,1,1]
 => [[1,2,3],[4],[5]]
 => 7
[2,2,1]
 => [[1,2],[3,4],[5]]
 => 6
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 0
[6]
 => [[1,2,3,4,5,6]]
 => 15
[5,1]
 => [[1,2,3,4,5],[6]]
 => 14
[4,2]
 => [[1,2,3,4],[5,6]]
 => 13
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => 12
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 11
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => 9
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 9
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 8
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => 21
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => 20
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => 19
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 18
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => 18
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => 17
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 15
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 16
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 15
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => 14
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => 11
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 12
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => 10
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 0
[8]
 => [[1,2,3,4,5,6,7,8]]
 => 28
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => 27
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => 26
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => 25
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 25
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => 24
Description
The charge of a standard tableau.
Matching statistic: St001541
Mapping
St001541: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 0
[4]
 => 6
[3,1]
 => 5
[2,2]
 => 4
[2,1,1]
 => 3
[1,1,1,1]
 => 0
[5]
 => 10
[4,1]
 => 9
[3,2]
 => 8
[3,1,1]
 => 7
[2,2,1]
 => 6
[2,1,1,1]
 => 4
[1,1,1,1,1]
 => 0
[6]
 => 15
[5,1]
 => 14
[4,2]
 => 13
[4,1,1]
 => 12
[3,3]
 => 12
[3,2,1]
 => 11
[3,1,1,1]
 => 9
[2,2,2]
 => 9
[2,2,1,1]
 => 8
[2,1,1,1,1]
 => 5
[1,1,1,1,1,1]
 => 0
[7]
 => 21
[6,1]
 => 20
[5,2]
 => 19
[5,1,1]
 => 18
[4,3]
 => 18
[4,2,1]
 => 17
[4,1,1,1]
 => 15
[3,3,1]
 => 16
[3,2,2]
 => 15
[3,2,1,1]
 => 14
[3,1,1,1,1]
 => 11
[2,2,2,1]
 => 12
[2,2,1,1,1]
 => 10
[2,1,1,1,1,1]
 => 6
[1,1,1,1,1,1,1]
 => 0
[8]
 => 28
[7,1]
 => 27
[6,2]
 => 26
[6,1,1]
 => 25
[5,3]
 => 25
[5,2,1]
 => 24
[]
 => ? = 0
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St000059
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000059: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 99%
Values
[1]
 => [1]
 => [[1]]
 => 0
[2]
 => [1,1]
 => [[1],[2]]
 => 1
[1,1]
 => [2]
 => [[1,2]]
 => 0
[3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
[1,1,1]
 => [3]
 => [[1,2,3]]
 => 0
[4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 6
[3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 5
[2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 4
[2,1,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 3
[1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 0
[5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 10
[4,1]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 9
[3,2]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => 8
[3,1,1]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => 7
[2,2,1]
 => [3,2]
 => [[1,2,3],[4,5]]
 => 6
[2,1,1,1]
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 0
[6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 15
[5,1]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => 14
[4,2]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => 13
[4,1,1]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => 12
[3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 12
[3,2,1]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => 11
[3,1,1,1]
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => 9
[2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 9
[2,2,1,1]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 8
[2,1,1,1,1]
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 5
[1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 0
[7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 21
[6,1]
 => [2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 20
[5,2]
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => 19
[5,1,1]
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => 18
[4,3]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => 18
[4,2,1]
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => 17
[4,1,1,1]
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => 15
[3,3,1]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 16
[3,2,2]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => 15
[3,2,1,1]
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => 14
[3,1,1,1,1]
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => 11
[2,2,2,1]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 12
[2,2,1,1,1]
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 10
[2,1,1,1,1,1]
 => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => 6
[1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 28
[7,1]
 => [2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => 27
[6,2]
 => [2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => 26
[6,1,1]
 => [3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => 25
[5,3]
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => 25
[5,2,1]
 => [3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => 24
[12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 66
Description
The inversion number of a standard tableau as defined by Haglund and Stevens. Their inversion number is the total number of inversion pairs for the tableau. An inversion pair is defined as a pair of cells (a,b), (x,y) such that the content of (x,y) is greater than the content of (a,b) and (x,y) is north of the inversion path of (a,b), where the inversion path is defined in detail in [1].
Matching statistic: St000567
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 91%●distinct values known / distinct values provided: 83%
Values
[1]
 => [1]
 => ? = 0
[2]
 => [1,1]
 => 1
[1,1]
 => [2]
 => 0
[3]
 => [1,1,1]
 => 3
[2,1]
 => [2,1]
 => 2
[1,1,1]
 => [3]
 => 0
[4]
 => [1,1,1,1]
 => 6
[3,1]
 => [2,1,1]
 => 5
[2,2]
 => [2,2]
 => 4
[2,1,1]
 => [3,1]
 => 3
[1,1,1,1]
 => [4]
 => 0
[5]
 => [1,1,1,1,1]
 => 10
[4,1]
 => [2,1,1,1]
 => 9
[3,2]
 => [2,2,1]
 => 8
[3,1,1]
 => [3,1,1]
 => 7
[2,2,1]
 => [3,2]
 => 6
[2,1,1,1]
 => [4,1]
 => 4
[1,1,1,1,1]
 => [5]
 => 0
[6]
 => [1,1,1,1,1,1]
 => 15
[5,1]
 => [2,1,1,1,1]
 => 14
[4,2]
 => [2,2,1,1]
 => 13
[4,1,1]
 => [3,1,1,1]
 => 12
[3,3]
 => [2,2,2]
 => 12
[3,2,1]
 => [3,2,1]
 => 11
[3,1,1,1]
 => [4,1,1]
 => 9
[2,2,2]
 => [3,3]
 => 9
[2,2,1,1]
 => [4,2]
 => 8
[2,1,1,1,1]
 => [5,1]
 => 5
[1,1,1,1,1,1]
 => [6]
 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 21
[6,1]
 => [2,1,1,1,1,1]
 => 20
[5,2]
 => [2,2,1,1,1]
 => 19
[5,1,1]
 => [3,1,1,1,1]
 => 18
[4,3]
 => [2,2,2,1]
 => 18
[4,2,1]
 => [3,2,1,1]
 => 17
[4,1,1,1]
 => [4,1,1,1]
 => 15
[3,3,1]
 => [3,2,2]
 => 16
[3,2,2]
 => [3,3,1]
 => 15
[3,2,1,1]
 => [4,2,1]
 => 14
[3,1,1,1,1]
 => [5,1,1]
 => 11
[2,2,2,1]
 => [4,3]
 => 12
[2,2,1,1,1]
 => [5,2]
 => 10
[2,1,1,1,1,1]
 => [6,1]
 => 6
[1,1,1,1,1,1,1]
 => [7]
 => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => 28
[7,1]
 => [2,1,1,1,1,1,1]
 => 27
[6,2]
 => [2,2,1,1,1,1]
 => 26
[6,1,1]
 => [3,1,1,1,1,1]
 => 25
[5,3]
 => [2,2,2,1,1]
 => 25
[5,2,1]
 => [3,2,1,1,1]
 => 24
[5,1,1,1]
 => [4,1,1,1,1]
 => 22
[5,4,3,1]
 => [4,3,3,2,1]
 => ? = 65
[5,4,2,2]
 => [4,4,2,2,1]
 => ? = 64
[5,4,2,1,1]
 => [5,3,2,2,1]
 => ? = 63
[5,3,3,2]
 => [4,4,3,1,1]
 => ? = 63
[5,3,3,1,1]
 => [5,3,3,1,1]
 => ? = 62
[5,3,2,2,1]
 => [5,4,2,1,1]
 => ? = 61
[4,4,3,2]
 => [4,4,3,2]
 => ? = 62
[4,4,3,1,1]
 => [5,3,3,2]
 => ? = 61
[4,4,2,2,1]
 => [5,4,2,2]
 => ? = 60
[4,3,3,2,1]
 => [5,4,3,1]
 => ? = 59
[5,4,3,2]
 => [4,4,3,2,1]
 => ? = 75
[5,4,3,1,1]
 => [5,3,3,2,1]
 => ? = 74
[5,4,2,2,1]
 => [5,4,2,2,1]
 => ? = 73
[5,3,3,2,1]
 => [5,4,3,1,1]
 => ? = 72
[4,4,3,2,1]
 => [5,4,3,2]
 => ? = 71
[5,4,3,2,1]
 => [5,4,3,2,1]
 => ? = 85
[]
 => []
 => ? = 0
Description
The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2} for a partition \lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n, see [1]. This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St000169
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 66% ●values known / values provided: 74%●distinct values known / distinct values provided: 66%
Values
[1]
 => [[1]]
 => [[1]]
 => 0
[2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
[3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[2,1]
 => [[1,2],[3]]
 => [[1,3],[2]]
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
[4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 6
[3,1]
 => [[1,2,3],[4]]
 => [[1,4],[2],[3]]
 => 5
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [[1,3,4],[2]]
 => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 10
[4,1]
 => [[1,2,3,4],[5]]
 => [[1,5],[2],[3],[4]]
 => 9
[3,2]
 => [[1,2,3],[4,5]]
 => [[1,4],[2,5],[3]]
 => 8
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [[1,4,5],[2],[3]]
 => 7
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [[1,3,5],[2,4]]
 => 6
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[1,3,4,5],[2]]
 => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
[6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 15
[5,1]
 => [[1,2,3,4,5],[6]]
 => [[1,6],[2],[3],[4],[5]]
 => 14
[4,2]
 => [[1,2,3,4],[5,6]]
 => [[1,5],[2,6],[3],[4]]
 => 13
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [[1,5,6],[2],[3],[4]]
 => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 12
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [[1,4,6],[2,5],[3]]
 => 11
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [[1,4,5,6],[2],[3]]
 => 9
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 9
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [[1,3,5,6],[2,4]]
 => 8
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [[1,3,4,5,6],[2]]
 => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 21
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 20
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [[1,6],[2,7],[3],[4],[5]]
 => 19
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [[1,6,7],[2],[3],[4],[5]]
 => 18
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [[1,5],[2,6],[3,7],[4]]
 => 18
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [[1,5,7],[2,6],[3],[4]]
 => 17
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [[1,5,6,7],[2],[3],[4]]
 => 15
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [[1,4,7],[2,5],[3,6]]
 => 16
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [[1,4,6],[2,5,7],[3]]
 => 15
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [[1,4,6,7],[2,5],[3]]
 => 14
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [[1,4,5,6,7],[2],[3]]
 => 11
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [[1,3,5,7],[2,4,6]]
 => 12
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [[1,3,5,6,7],[2,4]]
 => 10
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [[1,3,4,5,6,7],[2]]
 => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 0
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 28
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [[1,8],[2],[3],[4],[5],[6],[7]]
 => 27
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [[1,7],[2,8],[3],[4],[5],[6]]
 => 26
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [[1,7,8],[2],[3],[4],[5],[6]]
 => 25
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [[1,6],[2,7],[3,8],[4],[5]]
 => 25
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [[1,6,8],[2,7],[3],[4],[5]]
 => 24
[5,4,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11]]
 => [[1,6,10],[2,7,11],[3,8],[4,9],[5]]
 => ? = 47
[5,4,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11]]
 => [[1,6,10,11],[2,7],[3,8],[4,9],[5]]
 => ? = 46
[5,3,3]
 => [[1,2,3,4,5],[6,7,8],[9,10,11]]
 => [[1,6,9],[2,7,10],[3,8,11],[4],[5]]
 => ? = 46
[5,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11]]
 => [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
 => ? = 45
[5,3,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
 => [[1,6,9,10,11],[2,7],[3,8],[4],[5]]
 => ? = 43
[5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [[1,6,8,10],[2,7,9,11],[3],[4],[5]]
 => ? = 43
[5,2,2,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
 => [[1,6,8,10,11],[2,7,9],[3],[4],[5]]
 => ? = 42
[4,4,3]
 => [[1,2,3,4],[5,6,7,8],[9,10,11]]
 => [[1,5,9],[2,6,10],[3,7,11],[4,8]]
 => ? = 45
[4,4,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11]]
 => [[1,5,9,11],[2,6,10],[3,7],[4,8]]
 => ? = 44
[4,4,1,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
 => [[1,5,9,10,11],[2,6],[3,7],[4,8]]
 => ? = 42
[4,3,3,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11]]
 => [[1,5,8,11],[2,6,9],[3,7,10],[4]]
 => ? = 43
[4,3,2,2]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11]]
 => [[1,5,8,10],[2,6,9,11],[3,7],[4]]
 => ? = 42
[4,3,2,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
 => [[1,5,8,10,11],[2,6,9],[3,7],[4]]
 => ? = 41
[4,2,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
 => [[1,5,7,9,11],[2,6,8,10],[3],[4]]
 => ? = 39
[3,3,3,2]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9]]
 => ? = 40
[3,3,3,1,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
 => [[1,4,7,10,11],[2,5,8],[3,6,9]]
 => ? = 39
[3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [[1,4,7,9,11],[2,5,8,10],[3,6]]
 => ? = 38
[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
[6,4,2]
 => [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
 => [[1,7,11],[2,8,12],[3,9],[4,10],[5],[6]]
 => ? = 58
[5,4,3]
 => [[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
[5,4,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
 => [[1,6,10,12],[2,7,11],[3,8],[4,9],[5]]
 => ? = 55
[5,4,1,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
 => [[1,6,10,11,12],[2,7],[3,8],[4,9],[5]]
 => ? = 53
[5,3,3,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
 => [[1,6,9,12],[2,7,10],[3,8,11],[4],[5]]
 => ? = 54
[5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => [[1,6,9,11],[2,7,10,12],[3,8],[4],[5]]
 => ? = 53
[5,3,2,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
 => [[1,6,9,11,12],[2,7,10],[3,8],[4],[5]]
 => ? = 52
[5,2,2,2,1]
 => [[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
[4,4,3,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
 => [[1,5,9,12],[2,6,10],[3,7,11],[4,8]]
 => ? = 53
[4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => [[1,5,9,11],[2,6,10,12],[3,7],[4,8]]
 => ? = 52
[4,4,2,1,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
 => [[1,5,9,11,12],[2,6,10],[3,7],[4,8]]
 => ? = 51
[4,3,3,2]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
 => [[1,5,8,11],[2,6,9,12],[3,7,10],[4]]
 => ? = 51
[4,3,3,1,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
 => [[1,5,8,11,12],[2,6,9],[3,7,10],[4]]
 => ? = 50
[4,3,2,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
 => [[1,5,8,10,12],[2,6,9,11],[3,7],[4]]
 => ? = 49
[3,3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
 => [[1,4,7,10,12],[2,5,8,11],[3,6,9]]
 => ? = 47
[3,3,2,2,1,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
 => [[1,4,7,9,11,12],[2,5,8,10],[3,6]]
 => ? = 44
[5,4,3,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
 => [[1,6,10,13],[2,7,11],[3,8,12],[4,9],[5]]
 => ? = 65
[5,4,2,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
 => [[1,6,10,12],[2,7,11,13],[3,8],[4,9],[5]]
 => ? = 64
[5,4,2,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
 => [[1,6,10,12,13],[2,7,11],[3,8],[4,9],[5]]
 => ? = 63
[5,3,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
 => [[1,6,9,12],[2,7,10,13],[3,8,11],[4],[5]]
 => ? = 63
[5,3,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
 => [[1,6,9,12,13],[2,7,10],[3,8,11],[4],[5]]
 => ? = 62
[5,3,2,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
 => [[1,6,9,11,13],[2,7,10,12],[3,8],[4],[5]]
 => ? = 61
[4,4,3,2]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
 => [[1,5,9,12],[2,6,10,13],[3,7,11],[4,8]]
 => ? = 62
[4,4,3,1,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
 => [[1,5,9,12,13],[2,6,10],[3,7,11],[4,8]]
 => ? = 61
[4,4,2,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
 => [[1,5,9,11,13],[2,6,10,12],[3,7],[4,8]]
 => ? = 60
[4,3,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
 => [[1,5,8,11,13],[2,6,9,12],[3,7,10],[4]]
 => ? = 59
[5,4,3,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
 => [[1,6,10,13],[2,7,11,14],[3,8,12],[4,9],[5]]
 => ? = 75
[5,4,3,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
 => [[1,6,10,13,14],[2,7,11],[3,8,12],[4,9],[5]]
 => ? = 74
[5,4,2,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
 => [[1,6,10,12,14],[2,7,11,13],[3,8],[4,9],[5]]
 => ? = 73
[5,3,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
 => [[1,6,9,12,14],[2,7,10,13],[3,8,11],[4],[5]]
 => ? = 72
[4,4,3,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
 => [[1,5,9,12,14],[2,6,10,13],[3,7,11],[4,8]]
 => ? = 71
[5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [[1,6,10,13,15],[2,7,11,14],[3,8,12],[4,9],[5]]
 => ? = 85
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
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 66% ●values known / values provided: 74%●distinct values known / distinct values provided: 66%
Values
[1]
 => [[1]]
 => [[1]]
 => 0
[2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
[3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 3
[2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
[4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 6
[3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 5
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 4
[2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 10
[4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 9
[3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 8
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 7
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 6
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
[6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 15
[5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 14
[4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 13
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 12
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 11
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 9
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 9
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 8
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 21
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 20
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 19
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => 18
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 18
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7]]
 => 17
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => 15
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 16
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 15
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6],[7]]
 => 14
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => 11
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 12
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 10
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 0
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 28
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => 27
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 26
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => 25
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7],[8]]
 => 25
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7],[8]]
 => 24
[5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => ? = 47
[5,4,1,1]
 => [[1,4,5,6,11],[2,8,9,10],[3],[7]]
 => [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
 => ? = 46
[5,3,3]
 => [[1,2,3,10,11],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
 => ? = 46
[5,3,2,1]
 => [[1,3,6,10,11],[2,5,9],[4,8],[7]]
 => [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
 => ? = 45
[5,3,1,1,1]
 => [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
 => [[1,2,3,4,7],[5,8],[6,9],[10],[11]]
 => ? = 43
[5,2,2,2]
 => [[1,2,9,10,11],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
 => ? = 43
[5,2,2,1,1]
 => [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
 => [[1,2,3,5,7],[4,6,8],[9],[10],[11]]
 => ? = 42
[4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11]]
 => ? = 45
[4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11]]
 => ? = 44
[4,4,1,1,1]
 => [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
 => [[1,2,3,4,8],[5,9],[6,10],[7,11]]
 => ? = 42
[4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11]]
 => ? = 43
[4,3,2,2]
 => [[1,2,7,11],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11]]
 => ? = 42
[4,3,2,1,1]
 => [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11]]
 => ? = 41
[4,2,2,2,1]
 => [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9],[10],[11]]
 => ? = 39
[3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 40
[3,3,3,1,1]
 => [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11]]
 => ? = 39
[3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11]]
 => ? = 38
[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
[6,4,2]
 => [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
 => ? = 58
[5,4,3]
 => [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
 => ? = 56
[5,4,2,1]
 => [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
 => ? = 55
[5,4,1,1,1]
 => [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
 => [[1,2,3,4,8],[5,9],[6,10],[7,11],[12]]
 => ? = 53
[5,3,3,1]
 => [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11],[12]]
 => ? = 54
[5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
 => ? = 53
[5,3,2,1,1]
 => [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11],[12]]
 => ? = 52
[5,2,2,2,1]
 => [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9],[10],[11],[12]]
 => ? = 50
[4,4,3,1]
 => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
 => [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
 => ? = 53
[4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
 => ? = 52
[4,4,2,1,1]
 => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
 => [[1,2,3,5,9],[4,6,10],[7,11],[8,12]]
 => ? = 51
[4,3,3,2]
 => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
 => ? = 51
[4,3,3,1,1]
 => [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11],[12]]
 => ? = 50
[4,3,2,2,1]
 => [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
 => ? = 49
[3,3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
 => ? = 47
[3,3,2,2,1,1]
 => [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
 => [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
 => ? = 44
[5,4,3,1]
 => [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
 => [[1,2,5,9],[3,6,10],[4,7,11],[8,12],[13]]
 => ? = 65
[5,4,2,2]
 => [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
 => [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
 => ? = 64
[5,4,2,1,1]
 => [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
 => [[1,2,3,5,9],[4,6,10],[7,11],[8,12],[13]]
 => ? = 63
[5,3,3,2]
 => [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11],[12],[13]]
 => ? = 63
[5,3,3,1,1]
 => [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11],[12],[13]]
 => ? = 62
[5,3,2,2,1]
 => [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11],[12],[13]]
 => ? = 61
[4,4,3,2]
 => [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
 => [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13]]
 => ? = 62
[4,4,3,1,1]
 => [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
 => [[1,2,3,6,10],[4,7,11],[5,8,12],[9,13]]
 => ? = 61
[4,4,2,2,1]
 => [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
 => [[1,2,4,6,10],[3,5,7,11],[8,12],[9,13]]
 => ? = 60
[4,3,3,2,1]
 => [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13]]
 => ? = 59
[5,4,3,2]
 => [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
 => [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13],[14]]
 => ? = 75
[5,4,3,1,1]
 => [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
 => [[1,2,3,6,10],[4,7,11],[5,8,12],[9,13],[14]]
 => ? = 74
[5,4,2,2,1]
 => [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
 => [[1,2,4,6,10],[3,5,7,11],[8,12],[9,13],[14]]
 => ? = 73
[5,3,3,2,1]
 => [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13],[14]]
 => ? = 72
[4,4,3,2,1]
 => [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14]]
 => ? = 71
[5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
 => ? = 85
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: St000018
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 74%●distinct values known / distinct values provided: 66%
Values
[1]
 => [1]
 => [[1]]
 => [1] => 0
[2]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,1]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[3]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 3
[2,1]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,1,1]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 6
[3,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 5
[2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 4
[2,1,1]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 3
[1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 10
[4,1]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 9
[3,2]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 8
[3,1,1]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 7
[2,2,1]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 6
[2,1,1,1]
 => [4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 4
[1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 0
[6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 15
[5,1]
 => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 14
[4,2]
 => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 13
[4,1,1]
 => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 12
[3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 12
[3,2,1]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => 11
[3,1,1,1]
 => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => 9
[2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 9
[2,2,1,1]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => 8
[2,1,1,1,1]
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 5
[1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 0
[7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => 21
[6,1]
 => [2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => 20
[5,2]
 => [2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => 19
[5,1,1]
 => [3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => 18
[4,3]
 => [2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => 18
[4,2,1]
 => [3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => 17
[4,1,1,1]
 => [4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => 15
[3,3,1]
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => 16
[3,2,2]
 => [3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => 15
[3,2,1,1]
 => [4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => 14
[3,1,1,1,1]
 => [5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => 11
[2,2,2,1]
 => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => 12
[2,2,1,1,1]
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => 10
[2,1,1,1,1,1]
 => [6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => 6
[1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => 0
[8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => 28
[7,1]
 => [2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => 27
[6,2]
 => [2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => 26
[6,1,1]
 => [3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => 25
[5,3]
 => [2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => 25
[5,2,1]
 => [3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => 24
[5,4,2]
 => [3,3,2,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,4,5,6,1,2,3] => ? = 47
[5,4,1,1]
 => [4,2,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
 => [11,9,10,7,8,5,6,1,2,3,4] => ? = 46
[5,3,3]
 => [3,3,3,1,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
 => [11,10,7,8,9,4,5,6,1,2,3] => ? = 46
[5,3,2,1]
 => [4,3,2,1,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
 => [11,10,8,9,5,6,7,1,2,3,4] => ? = 45
[5,3,1,1,1]
 => [5,2,2,1,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
 => [11,10,8,9,6,7,1,2,3,4,5] => ? = 43
[5,2,2,2]
 => [4,4,1,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
 => [11,10,9,5,6,7,8,1,2,3,4] => ? = 43
[5,2,2,1,1]
 => [5,3,1,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
 => [11,10,9,6,7,8,1,2,3,4,5] => ? = 42
[4,4,3]
 => [3,3,3,2]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11]]
 => [10,11,7,8,9,4,5,6,1,2,3] => ? = 45
[4,4,2,1]
 => [4,3,2,2]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11]]
 => [10,11,8,9,5,6,7,1,2,3,4] => ? = 44
[4,4,1,1,1]
 => [5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [10,11,8,9,6,7,1,2,3,4,5] => ? = 42
[4,3,3,1]
 => [4,3,3,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11]]
 => [11,8,9,10,5,6,7,1,2,3,4] => ? = 43
[4,3,2,2]
 => [4,4,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11]]
 => [11,9,10,5,6,7,8,1,2,3,4] => ? = 42
[4,3,2,1,1]
 => [5,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11]]
 => [11,9,10,6,7,8,1,2,3,4,5] => ? = 41
[4,2,2,2,1]
 => [5,4,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11]]
 => [11,10,6,7,8,9,1,2,3,4,5] => ? = 39
[3,3,3,2]
 => [4,4,3]
 => [[1,2,3,4],[5,6,7,8],[9,10,11]]
 => [9,10,11,5,6,7,8,1,2,3,4] => ? = 40
[3,3,3,1,1]
 => [5,3,3]
 => [[1,2,3,4,5],[6,7,8],[9,10,11]]
 => [9,10,11,6,7,8,1,2,3,4,5] => ? = 39
[3,3,2,2,1]
 => [5,4,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11]]
 => [10,11,6,7,8,9,1,2,3,4,5] => ? = 38
[6,6]
 => [2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 60
[6,4,2]
 => [3,3,2,2,1,1]
 => [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]
 => [12,11,9,10,7,8,4,5,6,1,2,3] => ? = 58
[5,4,3]
 => [3,3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
 => [12,10,11,7,8,9,4,5,6,1,2,3] => ? = 56
[5,4,2,1]
 => [4,3,2,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
 => [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 55
[5,4,1,1,1]
 => [5,2,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
 => [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 53
[5,3,3,1]
 => [4,3,3,1,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
 => [12,11,8,9,10,5,6,7,1,2,3,4] => ? = 54
[5,3,2,2]
 => [4,4,2,1,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
 => [12,11,9,10,5,6,7,8,1,2,3,4] => ? = 53
[5,3,2,1,1]
 => [5,3,2,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
 => [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 52
[5,2,2,2,1]
 => [5,4,1,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
 => [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 50
[4,4,3,1]
 => [4,3,3,2]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12]]
 => [11,12,8,9,10,5,6,7,1,2,3,4] => ? = 53
[4,4,2,2]
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 52
[4,4,2,1,1]
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => [11,12,9,10,6,7,8,1,2,3,4,5] => ? = 51
[4,3,3,2]
 => [4,4,3,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
 => [12,9,10,11,5,6,7,8,1,2,3,4] => ? = 51
[4,3,3,1,1]
 => [5,3,3,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12]]
 => [12,9,10,11,6,7,8,1,2,3,4,5] => ? = 50
[4,3,2,2,1]
 => [5,4,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
 => [12,10,11,6,7,8,9,1,2,3,4,5] => ? = 49
[3,3,3,2,1]
 => [5,4,3]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12]]
 => [10,11,12,6,7,8,9,1,2,3,4,5] => ? = 47
[3,3,2,2,1,1]
 => [6,4,2]
 => [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
 => [11,12,7,8,9,10,1,2,3,4,5,6] => ? = 44
[5,4,3,1]
 => [4,3,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
 => [13,11,12,8,9,10,5,6,7,1,2,3,4] => ? = 65
[5,4,2,2]
 => [4,4,2,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
 => [13,11,12,9,10,5,6,7,8,1,2,3,4] => ? = 64
[5,4,2,1,1]
 => [5,3,2,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
 => [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 63
[5,3,3,2]
 => [4,4,3,1,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
 => [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 63
[5,3,3,1,1]
 => [5,3,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
 => [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 62
[5,3,2,2,1]
 => [5,4,2,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
 => [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 61
[4,4,3,2]
 => [4,4,3,2]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13]]
 => [12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 62
[4,4,3,1,1]
 => [5,3,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13]]
 => [12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 61
[4,4,2,2,1]
 => [5,4,2,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13]]
 => [12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 60
[4,3,3,2,1]
 => [5,4,3,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13]]
 => [13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 59
[5,4,3,2]
 => [4,4,3,2,1]
 => [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
 => [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 75
[5,4,3,1,1]
 => [5,3,3,2,1]
 => [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
 => [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 74
[5,4,2,2,1]
 => [5,4,2,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
 => [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 73
[5,3,3,2,1]
 => [5,4,3,1,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
 => [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 72
[4,4,3,2,1]
 => [5,4,3,2]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
 => [13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 71
[5,4,3,2,1]
 => [5,4,3,2,1]
 => [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
 => [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 85
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions (i,i+1) needed to write \pi. Thus, it is also the Coxeter length of \pi.
Matching statistic: St000391
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 65% ●values known / values provided: 73%●distinct values known / distinct values provided: 65%
Values
[1]
 => [[1]]
 => [[1]]
 => => ? = 0
[2]
 => [[1,2]]
 => [[1],[2]]
 => 1 => 1
[1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0 => 0
[3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 11 => 3
[2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 01 => 2
[1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 00 => 0
[4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 111 => 6
[3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 011 => 5
[2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 101 => 4
[2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 001 => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 000 => 0
[5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 1111 => 10
[4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 0111 => 9
[3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 1011 => 8
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 0011 => 7
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 0101 => 6
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 0001 => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0000 => 0
[6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 11111 => 15
[5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 01111 => 14
[4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 10111 => 13
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 00111 => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 11011 => 12
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 01011 => 11
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 00011 => 9
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 10101 => 9
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 00101 => 8
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 00001 => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 00000 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 111111 => 21
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 011111 => 20
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 101111 => 19
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => 001111 => 18
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 110111 => 18
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7]]
 => 010111 => 17
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => 000111 => 15
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 011011 => 16
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 101011 => 15
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6],[7]]
 => 001011 => 14
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => 000011 => 11
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 010101 => 12
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 000101 => 10
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => 000001 => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [[1,2,3,4,5,6,7]]
 => 000000 => 0
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 1111111 => 28
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => 0111111 => 27
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [[1,3],[2,4],[5],[6],[7],[8]]
 => 1011111 => 26
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => 0011111 => 25
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7],[8]]
 => 1101111 => 25
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7],[8]]
 => 0101111 => 24
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => 0001111 => 22
[5,4,2]
 => [[1,2,5,6,11],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
 => 1011101111 => ? = 47
[5,4,1,1]
 => [[1,4,5,6,11],[2,8,9,10],[3],[7]]
 => [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
 => 0011101111 => ? = 46
[5,3,3]
 => [[1,2,3,10,11],[4,5,6],[7,8,9]]
 => [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
 => 1101101111 => ? = 46
[5,3,2,1]
 => [[1,3,6,10,11],[2,5,9],[4,8],[7]]
 => [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
 => 0101101111 => ? = 45
[5,2,2,2]
 => [[1,2,9,10,11],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
 => 1010101111 => ? = 43
[4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11]]
 => 1101110111 => ? = 45
[4,4,2,1]
 => [[1,3,6,7],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11]]
 => 0101110111 => ? = 44
[4,3,3,1]
 => [[1,3,4,11],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11]]
 => 0110110111 => ? = 43
[4,3,2,2]
 => [[1,2,7,11],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11]]
 => 1010110111 => ? = 42
[4,3,2,1,1]
 => [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11]]
 => 0010110111 => ? = 41
[4,2,2,2,1]
 => [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9],[10],[11]]
 => 0101010111 => ? = 39
[3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => 1011011011 => ? = 40
[3,3,3,1,1]
 => [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11]]
 => 0011011011 => ? = 39
[3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11]]
 => 0101011011 => ? = 38
[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]]
 => 11111111111 => ? = 66
[6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
 => 11111011111 => ? = 60
[6,4,2]
 => [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
 => [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
 => 10111011111 => ? = 58
[5,4,3]
 => [[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 => ? = 56
[5,4,2,1]
 => [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
 => [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
 => 01011101111 => ? = 55
[5,4,1,1,1]
 => [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
 => [[1,2,3,4,8],[5,9],[6,10],[7,11],[12]]
 => 00011101111 => ? = 53
[5,3,3,1]
 => [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
 => [[1,2,5,8],[3,6,9],[4,7,10],[11],[12]]
 => 01101101111 => ? = 54
[5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
 => 10101101111 => ? = 53
[5,3,2,1,1]
 => [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
 => [[1,2,3,5,8],[4,6,9],[7,10],[11],[12]]
 => 00101101111 => ? = 52
[5,2,2,2,1]
 => [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9],[10],[11],[12]]
 => 01010101111 => ? = 50
[4,4,3,1]
 => [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
 => [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
 => 01101110111 => ? = 53
[4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
 => 10101110111 => ? = 52
[4,4,2,1,1]
 => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
 => [[1,2,3,5,9],[4,6,10],[7,11],[8,12]]
 => 00101110111 => ? = 51
[4,3,3,2]
 => [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
 => 10110110111 => ? = 51
[4,3,3,1,1]
 => [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11],[12]]
 => 00110110111 => ? = 50
[4,3,2,2,1]
 => [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
 => 01010110111 => ? = 49
[3,3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
 => 01011011011 => ? = 47
[3,3,2,2,1,1]
 => [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
 => [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
 => 00101011011 => ? = 44
[2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 10101010101 => ? = 36
[5,4,3,1]
 => [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
 => [[1,2,5,9],[3,6,10],[4,7,11],[8,12],[13]]
 => 011011101111 => ? = 65
[5,4,2,2]
 => [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
 => [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
 => 101011101111 => ? = 64
[5,4,2,1,1]
 => [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
 => [[1,2,3,5,9],[4,6,10],[7,11],[8,12],[13]]
 => 001011101111 => ? = 63
[5,3,3,2]
 => [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11],[12],[13]]
 => 101101101111 => ? = 63
[5,3,3,1,1]
 => [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
 => [[1,2,3,6,9],[4,7,10],[5,8,11],[12],[13]]
 => 001101101111 => ? = 62
[5,3,2,2,1]
 => [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11],[12],[13]]
 => 010101101111 => ? = 61
[4,4,3,2]
 => [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
 => [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13]]
 => 101101110111 => ? = 62
[4,4,3,1,1]
 => [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
 => [[1,2,3,6,10],[4,7,11],[5,8,12],[9,13]]
 => 001101110111 => ? = 61
[4,4,2,2,1]
 => [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
 => [[1,2,4,6,10],[3,5,7,11],[8,12],[9,13]]
 => 010101110111 => ? = 60
[4,3,3,2,1]
 => [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13]]
 => 010110110111 => ? = 59
[5,4,3,2]
 => [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
 => [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13],[14]]
 => 1011011101111 => ? = 75
[5,4,3,1,1]
 => [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
 => [[1,2,3,6,10],[4,7,11],[5,8,12],[9,13],[14]]
 => 0011011101111 => ? = 74
[5,4,2,2,1]
 => [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
 => [[1,2,4,6,10],[3,5,7,11],[8,12],[9,13],[14]]
 => 0101011101111 => ? = 73
[5,3,3,2,1]
 => [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
 => [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13],[14]]
 => 0101101101111 => ? = 72
[4,4,3,2,1]
 => [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14]]
 => 0101101110111 => ? = 71
[5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
 => [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
 => 01011011101111 => ? = 85
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000008
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 54%
Values
[1]
 => [[1]]
 => [1] => [1] => 0
[2]
 => [[1,2]]
 => [2] => [1,1] => 1
[1,1]
 => [[1],[2]]
 => [1,1] => [2] => 0
[3]
 => [[1,2,3]]
 => [3] => [1,1,1] => 3
[2,1]
 => [[1,2],[3]]
 => [2,1] => [2,1] => 2
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => [3] => 0
[4]
 => [[1,2,3,4]]
 => [4] => [1,1,1,1] => 6
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => [2,1,1] => 5
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => [1,2,1] => 4
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => [3,1] => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => [4] => 0
[5]
 => [[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => 10
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => [2,1,1,1] => 9
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => [1,2,1,1] => 8
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => [3,1,1] => 7
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => [2,2,1] => 6
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => [4,1] => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => [5] => 0
[6]
 => [[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1] => 15
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => [2,1,1,1,1] => 14
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => [1,2,1,1,1] => 13
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => [3,1,1,1] => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => [1,1,2,1,1] => 12
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => [2,2,1,1] => 11
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => [4,1,1] => 9
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => [1,2,2,1] => 9
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => [3,2,1] => 8
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => [5,1] => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => [6] => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => [1,1,1,1,1,1,1] => 21
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => [2,1,1,1,1,1] => 20
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => [1,2,1,1,1,1] => 19
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => [3,1,1,1,1] => 18
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => [1,1,2,1,1,1] => 18
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => [2,2,1,1,1] => 17
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => [4,1,1,1] => 15
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => [2,1,2,1,1] => 16
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => [1,2,2,1,1] => 15
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => [3,2,1,1] => 14
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => [5,1,1] => 11
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => [2,2,2,1] => 12
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => [4,2,1] => 10
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => [6,1] => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => [7] => 0
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [8] => [1,1,1,1,1,1,1,1] => 28
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [7,1] => [2,1,1,1,1,1,1] => 27
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [6,2] => [1,2,1,1,1,1,1] => 26
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [6,1,1] => [3,1,1,1,1,1] => 25
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [5,3] => [1,1,2,1,1,1,1] => 25
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [5,2,1] => [2,2,1,1,1,1] => 24
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [9] => [1,1,1,1,1,1,1,1,1] => ? = 36
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [8,1] => [2,1,1,1,1,1,1,1] => ? = 35
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [7,2] => [1,2,1,1,1,1,1,1] => ? = 34
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [7,1,1] => [3,1,1,1,1,1,1] => ? = 33
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [6,3] => [1,1,2,1,1,1,1,1] => ? = 33
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [6,2,1] => [2,2,1,1,1,1,1] => ? = 32
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [6,1,1,1] => [4,1,1,1,1,1] => ? = 30
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [5,4] => [1,1,1,2,1,1,1,1] => ? = 32
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [5,3,1] => [2,1,2,1,1,1,1] => ? = 31
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [5,2,2] => [1,2,2,1,1,1,1] => ? = 30
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [5,2,1,1] => [3,2,1,1,1,1] => ? = 29
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [5,1,1,1,1] => [5,1,1,1,1] => ? = 26
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [4,4,1] => [2,1,1,2,1,1,1] => ? = 30
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [4,3,2] => [1,2,1,2,1,1,1] => ? = 29
[4,3,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9]]
 => [4,3,1,1] => [3,1,2,1,1,1] => ? = 28
[4,2,2,1]
 => [[1,2,3,4],[5,6],[7,8],[9]]
 => [4,2,2,1] => [2,2,2,1,1,1] => ? = 27
[4,2,1,1,1]
 => [[1,2,3,4],[5,6],[7],[8],[9]]
 => [4,2,1,1,1] => [4,2,1,1,1] => ? = 25
[4,1,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8],[9]]
 => [4,1,1,1,1,1] => [6,1,1,1] => ? = 21
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [3,3,3] => [1,1,2,1,2,1,1] => ? = 27
[3,3,2,1]
 => [[1,2,3],[4,5,6],[7,8],[9]]
 => [3,3,2,1] => [2,2,1,2,1,1] => ? = 26
[3,3,1,1,1]
 => [[1,2,3],[4,5,6],[7],[8],[9]]
 => [3,3,1,1,1] => [4,1,2,1,1] => ? = 24
[3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => [3,2,2,2] => [1,2,2,2,1,1] => ? = 24
[3,2,2,1,1]
 => [[1,2,3],[4,5],[6,7],[8],[9]]
 => [3,2,2,1,1] => [3,2,2,1,1] => ? = 23
[3,2,1,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8],[9]]
 => [3,2,1,1,1,1] => [5,2,1,1] => ? = 20
[2,2,2,2,1]
 => [[1,2],[3,4],[5,6],[7,8],[9]]
 => [2,2,2,2,1] => [2,2,2,2,1] => ? = 20
[2,2,2,1,1,1]
 => [[1,2],[3,4],[5,6],[7],[8],[9]]
 => [2,2,2,1,1,1] => [4,2,2,1] => ? = 18
[1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => [1,1,1,1,1,1,1,1,1] => [9] => ? = 0
[9,1]
 => [[1,2,3,4,5,6,7,8,9],[10]]
 => [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
[8,2]
 => [[1,2,3,4,5,6,7,8],[9,10]]
 => [8,2] => [1,2,1,1,1,1,1,1,1] => ? = 43
[8,1,1]
 => [[1,2,3,4,5,6,7,8],[9],[10]]
 => [8,1,1] => [3,1,1,1,1,1,1,1] => ? = 42
[7,3]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => [7,3] => [1,1,2,1,1,1,1,1,1] => ? = 42
[7,1,1,1]
 => [[1,2,3,4,5,6,7],[8],[9],[10]]
 => [7,1,1,1] => [4,1,1,1,1,1,1] => ? = 39
[6,4]
 => [[1,2,3,4,5,6],[7,8,9,10]]
 => [6,4] => [1,1,1,2,1,1,1,1,1] => ? = 41
[6,3,1]
 => [[1,2,3,4,5,6],[7,8,9],[10]]
 => [6,3,1] => [2,1,2,1,1,1,1,1] => ? = 40
[6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => [6,2,2] => [1,2,2,1,1,1,1,1] => ? = 39
[6,2,1,1]
 => [[1,2,3,4,5,6],[7,8],[9],[10]]
 => [6,2,1,1] => [3,2,1,1,1,1,1] => ? = 38
[6,1,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9],[10]]
 => [6,1,1,1,1] => [5,1,1,1,1,1] => ? = 35
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [5,5] => [1,1,1,1,2,1,1,1,1] => ? = 40
[5,3,2]
 => [[1,2,3,4,5],[6,7,8],[9,10]]
 => [5,3,2] => [1,2,1,2,1,1,1,1] => ? = 38
[5,3,1,1]
 => [[1,2,3,4,5],[6,7,8],[9],[10]]
 => [5,3,1,1] => [3,1,2,1,1,1,1] => ? = 37
[5,2,2,1]
 => [[1,2,3,4,5],[6,7],[8,9],[10]]
 => [5,2,2,1] => [2,2,2,1,1,1,1] => ? = 36
[5,2,1,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9],[10]]
 => [5,2,1,1,1] => [4,2,1,1,1,1] => ? = 34
[5,1,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9],[10]]
 => [5,1,1,1,1,1] => [6,1,1,1,1] => ? = 30
[4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [4,4,2] => [1,2,1,1,2,1,1,1] => ? = 37
[4,4,1,1]
 => [[1,2,3,4],[5,6,7,8],[9],[10]]
 => [4,4,1,1] => [3,1,1,2,1,1,1] => ? = 36
[4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => [4,3,3] => [1,1,2,1,2,1,1,1] => ? = 36
[4,3,2,1]
 => [[1,2,3,4],[5,6,7],[8,9],[10]]
 => [4,3,2,1] => [2,2,1,2,1,1,1] => ? = 35
[4,3,1,1,1]
 => [[1,2,3,4],[5,6,7],[8],[9],[10]]
 => [4,3,1,1,1] => [4,1,2,1,1,1] => ? = 33
[4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [4,2,2,2] => [1,2,2,2,1,1,1] => ? = 33
[4,2,2,1,1]
 => [[1,2,3,4],[5,6],[7,8],[9],[10]]
 => [4,2,2,1,1] => [3,2,2,1,1,1] => ? = 32
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: St000246
(load all 4 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 51%
Values
[1]
 => [[1]]
 => [1] => [1] => 0
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 1
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 0
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 0
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 6
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 5
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [1,4,2,3] => 4
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 0
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => 10
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 9
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,4,2,3,5] => 8
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => 7
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,4,5,2,3] => 6
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => 4
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 15
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 14
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,4,2,3,5,6] => 13
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [1,2,6,3,4,5] => 12
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [1,4,5,2,3,6] => 11
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 9
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [1,6,2,5,3,4] => 9
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [1,5,4,6,2,3] => 8
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 5
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 21
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 20
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [1,4,2,3,5,6,7] => ? = 19
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => 18
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [1,2,6,3,4,5,7] => ? = 18
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [1,4,5,2,3,6,7] => ? = 17
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => 15
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => [1,4,2,7,3,5,6] => ? = 16
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [1,6,2,5,3,4,7] => ? = 15
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => [1,5,4,6,2,3,7] => ? = 14
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => 11
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [1,3,7,5,6,2,4] => ? = 12
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [1,6,5,4,7,2,3] => ? = 10
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => 6
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 0
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 28
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 27
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [1,4,2,3,5,6,7,8] => 26
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => 25
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => [1,2,6,3,4,5,7,8] => ? = 25
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => [1,4,5,2,3,6,7,8] => ? = 24
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => 22
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => 24
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8] => [1,4,2,7,3,5,6,8] => 23
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => [1,6,2,5,3,4,7,8] => 22
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => [1,5,4,6,2,3,7,8] => ? = 21
[4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => 18
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5] => [1,2,4,7,8,3,5,6] => ? = 21
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5] => [1,5,4,2,8,3,6,7] => 20
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8] => [1,3,7,5,6,2,4,8] => ? = 19
[3,2,1,1,1]
 => [[1,5,8],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8] => [1,6,5,4,7,2,3,8] => ? = 17
[3,1,1,1,1,1]
 => [[1,7,8],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => 13
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => 16
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4] => [1,3,7,8,5,6,2,4] => ? = 15
[2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6] => [1,7,6,5,4,8,2,3] => 12
[2,1,1,1,1,1,1]
 => [[1,8],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => 7
[7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => [3,4,1,2,5,6,7,8,9] => [1,4,2,3,5,6,7,8,9] => ? = 34
[6,3]
 => [[1,2,3,7,8,9],[4,5,6]]
 => [4,5,6,1,2,3,7,8,9] => [1,2,6,3,4,5,7,8,9] => ? = 33
[6,2,1]
 => [[1,3,6,7,8,9],[2,5],[4]]
 => [4,2,5,1,3,6,7,8,9] => [1,4,5,2,3,6,7,8,9] => ? = 32
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4,9] => [1,2,3,8,4,5,6,7,9] => ? = 32
[5,3,1]
 => [[1,3,4,8,9],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8,9] => [1,4,2,7,3,5,6,8,9] => ? = 31
[5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8,9] => [1,6,2,5,3,4,7,8,9] => ? = 30
[5,2,1,1]
 => [[1,4,7,8,9],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8,9] => [1,5,4,6,2,3,7,8,9] => ? = 29
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [6,2,7,8,9,1,3,4,5] => [1,4,2,3,9,5,6,7,8] => ? = 30
[4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5,9] => [1,2,4,7,8,3,5,6,9] => ? = 29
[4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5,9] => [1,5,4,2,8,3,6,7,9] => ? = 28
[4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8,9] => [1,3,7,5,6,2,4,8,9] => ? = 27
[4,2,1,1,1]
 => [[1,5,8,9],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8,9] => [1,6,5,4,7,2,3,8,9] => ? = 25
[3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => [1,2,9,3,4,8,5,6,7] => ? = 27
[3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => [7,4,8,2,5,9,1,3,6] => [1,3,2,7,8,9,4,5,6] => ? = 26
[3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => [7,4,3,2,8,9,1,5,6] => [1,6,5,4,2,9,3,7,8] => ? = 24
[3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2,9] => [1,8,2,7,3,6,4,5,9] => ? = 24
[3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4,9] => [1,3,7,8,5,6,2,4,9] => ? = 23
[3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6,9] => [1,7,6,5,4,8,2,3,9] => ? = 20
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [8,6,9,4,7,2,5,1,3] => [1,3,9,4,8,6,7,2,5] => ? = 20
[2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4],[6],[8]]
 => [8,6,4,3,9,2,7,1,5] => [1,3,8,7,9,5,6,2,4] => ? = 18
[2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3],[4],[5],[6],[8]]
 => [8,6,5,4,3,2,9,1,7] => [1,8,7,6,5,4,9,2,3] => ? = 14
[8,2]
 => [[1,2,5,6,7,8,9,10],[3,4]]
 => [3,4,1,2,5,6,7,8,9,10] => [1,4,2,3,5,6,7,8,9,10] => ? = 43
[7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => [4,5,6,1,2,3,7,8,9,10] => [1,2,6,3,4,5,7,8,9,10] => ? = 42
[7,2,1]
 => [[1,3,6,7,8,9,10],[2,5],[4]]
 => [4,2,5,1,3,6,7,8,9,10] => [1,4,5,2,3,6,7,8,9,10] => ? = 41
[6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4,9,10] => [1,2,3,8,4,5,6,7,9,10] => ? = 41
[6,3,1]
 => [[1,3,4,8,9,10],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8,9,10] => [1,4,2,7,3,5,6,8,9,10] => ? = 40
[6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8,9,10] => [1,6,2,5,3,4,7,8,9,10] => ? = 39
[6,2,1,1]
 => [[1,4,7,8,9,10],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8,9,10] => [1,5,4,6,2,3,7,8,9,10] => ? = 38
[5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,10,5,6,7,8,9] => ? = 40
[5,4,1]
 => [[1,3,4,5,10],[2,7,8,9],[6]]
 => [6,2,7,8,9,1,3,4,5,10] => [1,4,2,3,9,5,6,7,8,10] => ? = 39
[5,3,2]
 => [[1,2,5,9,10],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5,9,10] => [1,2,4,7,8,3,5,6,9,10] => ? = 38
[5,3,1,1]
 => [[1,4,5,9,10],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5,9,10] => [1,5,4,2,8,3,6,7,9,10] => ? = 37
[5,2,2,1]
 => [[1,3,8,9,10],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8,9,10] => [1,3,7,5,6,2,4,8,9,10] => ? = 36
[5,2,1,1,1]
 => [[1,5,8,9,10],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8,9,10] => [1,6,5,4,7,2,3,8,9,10] => ? = 34
[4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => [7,8,3,4,9,10,1,2,5,6] => [1,2,4,7,3,10,5,6,8,9] => ? = 37
Description
The number of non-inversions of a permutation. For a permutation of \{1,\ldots,n\}, this is given by \operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi).
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001094The depth index of a set partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St001759The Rajchgot index of a permutation. St000794The mak of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000004The major index of a permutation. St000446The disorder of a permutation. St000304The load of a permutation. St000154The sum of the descent bottoms of a permutation. St000305The inverse major index of a permutation. St000472The sum of the ascent bottoms of a permutation. St000833The comajor index of a permutation.