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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000566
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [2,2]
=> [2]
=> 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 0
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [4,1,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [4,1,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000017
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 96%●distinct values known / distinct values provided: 86%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,3,1,4,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,2,1,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [4,1,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [4,1,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,4,1,2,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1
[[0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,-1,1,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1]]
=> [1,8,2,3,4,5,6,7,9] => [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,-1,1],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,9,3,4,5,6,7,8] => [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
[[0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,-1,1,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> [1,9,2,3,4,5,6,7,8,10] => [8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> ? = 0
[[0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,-1,1,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1]]
=> [2,1,8,3,4,5,6,7,9] => [6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> ? = 1
[[0,0,0,0,0,1,0,0,0],[1,0,0,0,0,-1,0,1,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1]]
=> [1,8,2,3,4,5,6,7,9] => [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,-1,0,1],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,9,3,4,5,6,7,8] => [7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,-1,1],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,10,3,4,5,6,7,8,9] => [8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,9,8] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,6,9,8,7] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6,9,8] => [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,3,2,4,5,6,7,9,8] => [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,7,9,8] => [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[[1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[0,1,0,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [8,7,6,5,4,3,2,1,9] => [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0]]
=> [1,9,8,7,6,5,4,3,2] => [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> [9,8,7,6,5,4,3,2,1,10] => [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 3
[[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [1,8,7,6,5,4,3,2,9] => [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 2
[[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0]]
=> [1,2,9,8,7,6,5,4,3] => [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 2
[[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0]]
=> [1,10,9,8,7,6,5,4,3,2] => [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 3
[[0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1]]
=> [10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 4
[[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> [1,9,8,7,6,5,4,3,2,10] => [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 3
[[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0,0,0],[0,1,0,-1,1,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [2,1,8,7,6,5,4,3,9] => [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 3
[[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,-1,0,1,0],[0,0,0,1,-1,0,1,0,0],[0,0,1,-1,0,1,0,0,0],[0,1,-1,0,1,0,0,0,0],[1,-1,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> [1,8,7,6,5,4,3,2,9] => [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 2
[[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,-1,0,1],[0,0,0,0,1,-1,0,1,0],[0,0,0,1,-1,0,1,0,0],[0,0,1,-1,0,1,0,0,0],[0,1,-1,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0]]
=> [1,2,9,8,7,6,5,4,3] => [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 2
[[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0]]
=> [1,2,10,9,8,7,6,5,4,3] => [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 3
[[1,0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0]]
=> [1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 4
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,9,7,8] => [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6,9,8] => [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [1,3,2,4,5,6,7,9,8] => [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,7,9,8] => [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 1
[[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> [1,3,2,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7,8,9] => [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
Description
The number of inversions of a standard tableau.
Let $T$ be a tableau. An inversion is an attacking pair $(c,d)$ of the shape of $T$ (see [[St000016]] for a definition of this) such that the entry of $c$ in $T$ is greater than the entry of $d$.
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