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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001177
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: St000825
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000825: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 22%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000825: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 22%
Values
[1]
=> [[1]]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [[1],[2]]
=> [2,1] => 2
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 0
[3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> [3,2,1] => 6
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 3
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 0
[4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 12
[3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 8
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 6
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 20
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 15
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 12
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 10
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 8
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 30
[5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 24
[4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => 20
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 18
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 18
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => 15
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 12
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 12
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => 10
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[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]]
=> [7,6,5,4,3,2,1] => 42
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 35
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> [7,6,5,2,4,1,3] => ? ∊ {12,15,18,20,22,24,27,30}
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 28
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7]]
=> [7,3,6,2,5,1,4] => ? ∊ {12,15,18,20,22,24,27,30}
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => ? ∊ {12,15,18,20,22,24,27,30}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => 21
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ? ∊ {12,15,18,20,22,24,27,30}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => ? ∊ {12,15,18,20,22,24,27,30}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? ∊ {12,15,18,20,22,24,27,30}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => 14
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? ∊ {12,15,18,20,22,24,27,30}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? ∊ {12,15,18,20,22,24,27,30}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[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]]
=> [8,7,6,5,4,3,2,1] => 56
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => 48
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> [8,7,6,5,2,4,1,3] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => 40
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> [8,7,3,6,2,5,1,4] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8]]
=> [8,7,6,3,5,1,2,4] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => 32
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8]]
=> [8,4,7,3,6,1,2,5] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> [8,7,2,4,6,1,3,5] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8]]
=> [8,7,4,6,1,2,3,5] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => 24
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8]]
=> [5,8,2,4,7,1,3,6] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [[1,2,3,6],[4,7],[5,8]]
=> [5,8,4,7,1,2,3,6] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7],[8]]
=> [8,3,5,7,1,2,4,6] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7],[8]]
=> [8,5,7,1,2,3,4,6] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => 16
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => 20
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8]]
=> [4,6,8,1,2,3,5,7] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => ? ∊ {14,18,21,24,26,28,28,30,32,35,36,38,42}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => 8
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => 72
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => 63
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,2,4,1,3] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => 54
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> [9,8,7,3,6,2,5,1,4] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8],[9]]
=> [9,8,7,6,3,5,1,2,4] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => 45
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> [9,4,8,3,7,2,6,1,5] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8],[9]]
=> [9,8,4,7,3,6,1,2,5] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> [9,8,7,2,4,6,1,3,5] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8],[9]]
=> [9,8,7,4,6,1,2,3,5] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => 36
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [[1,2,6],[3,7],[4,8],[5,9]]
=> [5,9,4,8,3,7,1,2,6] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8],[9]]
=> [9,5,8,2,4,7,1,3,6] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [[1,2,3,6],[4,7],[5,8],[9]]
=> [9,5,8,4,7,1,2,3,6] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7],[8],[9]]
=> [9,8,3,5,7,1,2,4,6] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7],[8],[9]]
=> [9,8,5,7,1,2,3,4,6] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> [3,6,9,2,5,8,1,4,7] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9]]
=> [6,9,3,5,8,1,2,4,7] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [[1,2,3,4,7],[5,8],[6,9]]
=> [6,9,5,8,1,2,3,4,7] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> [9,2,4,6,8,1,3,5,7] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8],[9]]
=> [9,4,6,8,1,2,3,5,7] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8],[9]]
=> [9,6,8,1,2,3,4,5,7] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9]]
=> [3,5,7,9,1,2,4,6,8] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [[1,2,3,4,6,8],[5,7,9]]
=> [5,7,9,1,2,3,4,6,8] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [[1,2,3,4,5,6,8],[7,9]]
=> [7,9,1,2,3,4,5,6,8] => ? ∊ {16,21,24,24,28,30,30,32,33,35,36,37,39,40,42,42,44,48,48,51,56}
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,2,4,1,3] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10]]
=> [10,9,8,7,3,6,2,5,1,4] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,3,5,1,2,4] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9],[10]]
=> [10,9,4,8,3,7,2,6,1,5] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8],[9],[10]]
=> [10,9,8,4,7,3,6,1,2,5] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> [10,9,8,7,2,4,6,1,3,5] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[6,2,1,1]
=> [[1,4,7,8,9,10],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8],[9],[10]]
=> [10,9,8,7,4,6,1,2,3,5] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> [5,10,4,9,3,8,2,7,1,6] => ? ∊ {18,24,27,28,32,34,35,36,38,40,40,42,42,42,45,45,48,48,48,50,50,52,54,55,56,58,60,62,63,66,72}
Description
The sum of the major and the inverse major index of a permutation.
This statistic is the sum of [[St000004]] and [[St000305]].
Matching statistic: St001379
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001379: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 11%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001379: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 11%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 3
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 12
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 8
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 6
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 4
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 20
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 15
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 10
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 8
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 5
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 30
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 24
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => 20
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,6,5,4,3] => 18
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 18
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,3,2,6,5,4] => 15
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,3,6,5,4] => 12
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 12
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,2,4,3,6,5] => 10
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,2,3,4,6,5] => 6
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? ∊ {20,27,30,35,42}
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? ∊ {20,27,30,35,42}
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => ? ∊ {20,27,30,35,42}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => 28
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? ∊ {20,27,30,35,42}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,3,2,7,6,5,4] => 24
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 21
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,4,3,2,7,6,5] => 22
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [2,1,4,3,7,6,5] => ? ∊ {20,27,30,35,42}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => 18
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => 14
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => 15
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => 12
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => 7
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [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] => [8,7,6,5,4,3,2,1] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [2,1,8,7,6,5,4,3] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,8,7,6,5,4,3] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [3,2,1,8,7,6,5,4] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,3,2,8,7,6,5,4] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,8,7,6,5,4] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,4,3,2,8,7,6,5] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [2,1,4,3,8,7,6,5] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,4,3,8,7,6,5] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,4,8,7,6,5] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [2,1,5,4,3,8,7,6] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [1,2,5,4,3,8,7,6] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,3,2,5,4,8,7,6] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,2,3,5,4,8,7,6] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,3,4,5,8,7,6] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [1,2,4,3,6,5,8,7] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? ∊ {0,8,14,16,18,20,21,24,24,26,28,28,30,32,32,35,36,38,40,42,48,56}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [2,1,9,8,7,6,5,4,3] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,9,8,7,6,5,4,3] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [3,2,1,9,8,7,6,5,4] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,3,2,9,8,7,6,5,4] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,9,8,7,6,5,4] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [4,3,2,1,9,8,7,6,5] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,4,3,2,9,8,7,6,5] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [2,1,4,3,9,8,7,6,5] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,4,3,9,8,7,6,5] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [1,5,4,3,2,9,8,7,6] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [2,1,5,4,3,9,8,7,6] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,2,5,4,3,9,8,7,6] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,3,2,5,4,9,8,7,6] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,3,5,4,9,8,7,6] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,4,5,9,8,7,6] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [1,3,2,6,5,4,9,8,7] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [1,2,3,6,5,4,9,8,7] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [2,1,4,3,6,5,9,8,7] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,2,4,3,6,5,9,8,7] => ? ∊ {0,9,16,18,21,24,24,27,28,30,30,32,33,35,36,36,37,39,40,42,42,44,45,48,48,51,54,56,63,72}
Description
The number of inversions plus the major index of a permutation.
This is, the sum of [[St000004]] and [[St000018]].
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