Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001177
Mapping
St001177: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 2
[3]
 => 0
[2,1]
 => 3
[1,1,1]
 => 6
[4]
 => 0
[3,1]
 => 4
[2,2]
 => 6
[2,1,1]
 => 8
[1,1,1,1]
 => 12
[5]
 => 0
[4,1]
 => 5
[3,2]
 => 8
[3,1,1]
 => 10
[2,2,1]
 => 12
[2,1,1,1]
 => 15
[1,1,1,1,1]
 => 20
[6]
 => 0
[5,1]
 => 6
[4,2]
 => 10
[4,1,1]
 => 12
[3,3]
 => 12
[3,2,1]
 => 15
[3,1,1,1]
 => 18
[2,2,2]
 => 18
[2,2,1,1]
 => 20
[2,1,1,1,1]
 => 24
[1,1,1,1,1,1]
 => 30
[7]
 => 0
[6,1]
 => 7
[5,2]
 => 12
[5,1,1]
 => 14
[4,3]
 => 15
[4,2,1]
 => 18
[4,1,1,1]
 => 21
[3,3,1]
 => 20
[3,2,2]
 => 22
[3,2,1,1]
 => 24
[3,1,1,1,1]
 => 28
[2,2,2,1]
 => 27
[2,2,1,1,1]
 => 30
[2,1,1,1,1,1]
 => 35
[1,1,1,1,1,1,1]
 => 42
[8]
 => 0
[7,1]
 => 8
[6,2]
 => 14
[6,1,1]
 => 16
[5,3]
 => 18
[5,2,1]
 => 21
Description
Twice the mean value of the major index among all standard Young tableaux of a partition. For a partition $\lambda$ of $n$, this mean value is given in [1, Proposition 3.1] by $$\frac{1}{2}\Big(\binom{n}{2} - \sum_i\binom{\lambda_i}{2} + \sum_i\binom{\lambda_i'}{2}\Big),$$ where $\lambda_i$ is the size of the $i$-th row of $\lambda$ and $\lambda_i'$ is the size of the $i$-th column.
Matching statistic: St001379
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001379: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 7%
Values
[1]
 => [[1]]
 => [1] => 0
[2]
 => [[1,2]]
 => [1,2] => 0
[1,1]
 => [[1],[2]]
 => [2,1] => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => 0
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => 3
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 6
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 4
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 6
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 8
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 12
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 0
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => 5
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 8
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 10
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 12
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 15
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 20
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 0
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 6
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => 10
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => 12
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 12
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => 15
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => 18
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 18
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => 20
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => 24
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 30
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => 0
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => ? = 7
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => ? = 12
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => ? = 14
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => ? = 15
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => ? = 18
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => ? = 21
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [7,4,5,6,1,2,3] => ? = 20
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => ? = 22
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => ? = 24
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => ? = 28
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => ? = 27
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => ? = 30
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => ? = 35
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ? = 42
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [1,2,3,4,5,6,7,8] => ? = 0
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [8,1,2,3,4,5,6,7] => ? = 8
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [7,8,1,2,3,4,5,6] => ? = 14
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [8,7,1,2,3,4,5,6] => ? = 16
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => ? = 18
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [8,6,7,1,2,3,4,5] => ? = 21
[5,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8]]
 => [8,7,6,1,2,3,4,5] => ? = 24
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 20
[4,3,1]
 => [[1,2,3,4],[5,6,7],[8]]
 => [8,5,6,7,1,2,3,4] => ? = 24
[4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => ? = 26
[4,2,1,1]
 => [[1,2,3,4],[5,6],[7],[8]]
 => [8,7,5,6,1,2,3,4] => ? = 28
[4,1,1,1,1]
 => [[1,2,3,4],[5],[6],[7],[8]]
 => [8,7,6,5,1,2,3,4] => ? = 32
[3,3,2]
 => [[1,2,3],[4,5,6],[7,8]]
 => [7,8,4,5,6,1,2,3] => ? = 28
[3,3,1,1]
 => [[1,2,3],[4,5,6],[7],[8]]
 => [8,7,4,5,6,1,2,3] => ? = 30
[3,2,2,1]
 => [[1,2,3],[4,5],[6,7],[8]]
 => [8,6,7,4,5,1,2,3] => ? = 32
[3,2,1,1,1]
 => [[1,2,3],[4,5],[6],[7],[8]]
 => [8,7,6,4,5,1,2,3] => ? = 35
[3,1,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,1,2,3] => ? = 40
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ? = 36
[2,2,2,1,1]
 => [[1,2],[3,4],[5,6],[7],[8]]
 => [8,7,5,6,3,4,1,2] => ? = 38
[2,2,1,1,1,1]
 => [[1,2],[3,4],[5],[6],[7],[8]]
 => [8,7,6,5,3,4,1,2] => ? = 42
[2,1,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,1,2] => ? = 48
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => ? = 56
[9]
 => [[1,2,3,4,5,6,7,8,9]]
 => [1,2,3,4,5,6,7,8,9] => ? = 0
[8,1]
 => [[1,2,3,4,5,6,7,8],[9]]
 => [9,1,2,3,4,5,6,7,8] => ? = 9
[7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => [8,9,1,2,3,4,5,6,7] => ? = 16
[7,1,1]
 => [[1,2,3,4,5,6,7],[8],[9]]
 => [9,8,1,2,3,4,5,6,7] => ? = 18
[6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => [7,8,9,1,2,3,4,5,6] => ? = 21
[6,2,1]
 => [[1,2,3,4,5,6],[7,8],[9]]
 => [9,7,8,1,2,3,4,5,6] => ? = 24
[6,1,1,1]
 => [[1,2,3,4,5,6],[7],[8],[9]]
 => [9,8,7,1,2,3,4,5,6] => ? = 27
[5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => [6,7,8,9,1,2,3,4,5] => ? = 24
[5,3,1]
 => [[1,2,3,4,5],[6,7,8],[9]]
 => [9,6,7,8,1,2,3,4,5] => ? = 28
[5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => ? = 30
[5,2,1,1]
 => [[1,2,3,4,5],[6,7],[8],[9]]
 => [9,8,6,7,1,2,3,4,5] => ? = 32
[5,1,1,1,1]
 => [[1,2,3,4,5],[6],[7],[8],[9]]
 => [9,8,7,6,1,2,3,4,5] => ? = 36
[4,4,1]
 => [[1,2,3,4],[5,6,7,8],[9]]
 => [9,5,6,7,8,1,2,3,4] => ? = 30
[4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => [8,9,5,6,7,1,2,3,4] => ? = 33
Description
The number of inversions plus the major index of a permutation. This is, the sum of [[St000004]] and [[St000018]].