Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000566
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00225: Semistandard tableaux weightInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[4,0],[2]]
=> [[1,1,2,2]]
=> [2,2]
=> [2]
=> 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [2,2]
=> [2]
=> 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [3,2]
=> [2]
=> 1
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [3,2]
=> [2]
=> 1
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,2]
=> [2]
=> 1
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [3,2]
=> [2]
=> 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [2,2]
=> [2]
=> 1
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [2,1,1]
=> [1,1]
=> 0
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [2,2]
=> [2]
=> 1
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [2,1,1]
=> [1,1]
=> 0
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [2,1,1]
=> [1,1]
=> 0
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [2,2]
=> [2]
=> 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [2,2]
=> [2]
=> 1
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [2,2]
=> [2]
=> 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,1,1]
=> [1,1]
=> 0
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,1,1]
=> [1,1]
=> 0
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [2,1,1]
=> [1,1]
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,1,1]
=> [1,1]
=> 0
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,1,1]
=> [1,1]
=> 0
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,1,1]
=> [1,1]
=> 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [1,1,1]
=> [1,1]
=> 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 0
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 $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00225: Semistandard tableaux weightInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 57% values known / values provided: 96%distinct values known / distinct values provided: 57%
Values
[[4,0],[2]]
=> [[1,1,2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[[4,3,2,1],[4,3,2],[4,3],[4]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,2],[4,3],[3]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,2],[4,2],[4]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,2],[4,2],[3]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 6
[[4,3,2,1],[4,3,2],[3,2],[3]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,2],[4,2],[2]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,2],[3,2],[2]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,3],[4]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,3],[3]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,2],[4]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[[4,3,2,1],[4,3,1],[4,2],[3]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[4,3,1],[3,2],[3]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[4,3,1],[4,2],[2]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[[4,3,2,1],[4,2,1],[4,2],[4]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[4,2],[3]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 6
[[4,3,2,1],[3,2,1],[3,2],[3]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[4,2],[2]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[3,2],[2]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[3,2,1],[3,2],[2]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,1],[4]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,1],[3]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[4,3,1],[3,1],[3]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,1],[2]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[4,3,1],[3,1],[2]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[[4,3,2,1],[4,2,1],[4,1],[4]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[4,1],[3]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[4,2,1],[3,1],[3]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 6
[[4,3,2,1],[3,2,1],[3,1],[3]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[4,1],[2]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[4,2,1],[3,1],[2]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 5
[[4,3,2,1],[3,2,1],[3,1],[2]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [4,2,2,2]
=> [[1,2,3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[[4,3,2,1],[4,2,1],[2,1],[2]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[3,2,1],[2,1],[2]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[4,1],[1]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,3,1],[3,1],[1]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[4,1],[1]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[3,1],[1]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 6
[[4,3,2,1],[3,2,1],[3,1],[1]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[4,2,1],[2,1],[1]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
[[4,3,2,1],[3,2,1],[2,1],[1]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> ? = 4
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 $d$.
Matching statistic: St001926
Mp00036: Gelfand-Tsetlin patterns to semistandard tableauSemistandard tableaux
Mp00075: Semistandard tableaux reading word permutationPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001926: Signed permutations ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 14%
Values
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ? = 1 + 3
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => ? = 1 + 3
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1 + 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 0 + 3
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 + 3
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 3
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 + 3
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ? = 1 + 3
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ? = 0 + 3
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ? = 1 + 3
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ? = 0 + 3
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ? = 0 + 3
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ? = 1 + 3
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0 + 3
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 1 + 3
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0 + 3
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => ? = 1 + 3
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ? = 0 + 3
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2,4] => ? = 0 + 3
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0 + 3
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => ? = 0 + 3
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => ? = 1 + 3
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1 + 3
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 0 + 3
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1 + 3
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => ? = 0 + 3
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => ? = 0 + 3
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => ? = 1 + 3
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ? = 0 + 3
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => ? = 0 + 3
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => ? = 0 + 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[2,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[1,1,1,0],[1,1,0],[1,0],[0]]
=> [[2],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 0 + 3
[[1,1,1,0],[1,1,0],[1,0],[1]]
=> [[1],[3],[4]]
=> [3,2,1] => [3,2,1] => 3 = 0 + 3
[[1,1,1,0],[1,1,0],[1,1],[1]]
=> [[1],[2],[4]]
=> [3,2,1] => [3,2,1] => 3 = 0 + 3
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3 = 0 + 3
[[6,0],[2]]
=> [[1,1,2,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 + 3
[[6,0],[3]]
=> [[1,1,1,2,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 3 + 3
[[6,0],[4]]
=> [[1,1,1,1,2,2]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 + 3
[[5,1],[2]]
=> [[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 1 + 3
[[5,1],[3]]
=> [[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 3 + 3
[[5,1],[4]]
=> [[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 1 + 3
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 1 + 3
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 3 + 3
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 1 + 3
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3 + 3
[[5,0,0],[2,0],[0]]
=> [[2,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[5,0,0],[2,0],[1]]
=> [[1,2,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0 + 3
[[5,0,0],[2,0],[2]]
=> [[1,1,3,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[5,0,0],[3,0],[0]]
=> [[2,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[5,0,0],[3,0],[1]]
=> [[1,2,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[5,0,0],[3,0],[2]]
=> [[1,1,2,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[5,0,0],[3,0],[3]]
=> [[1,1,1,3,3]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 + 3
[[3,0,0,0,0],[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,5]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,5]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,5]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,5]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,5]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,5]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3 = 0 + 3
[[2,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3,5],[4]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2,5],[4]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,5],[4]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,5],[3]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,5],[3]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1,5],[2]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4],[5]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4],[5]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[5]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[5]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[5]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[5]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[1,1,0],[1,1],[1]]
=> [[1,4],[2]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[2,0,0],[1,0],[0]]
=> [[2,3],[4]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[2,0,0],[1,0],[1]]
=> [[1,3],[4]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[2,0,0],[2,0],[1]]
=> [[1,2],[4]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 3 = 0 + 3
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 3 = 0 + 3
Description
Sparre Andersen's position of the maximum of a signed permutation. For $\pi$ a signed permutation of length $n$, first create the tuple $x = (x_1, \dots, x_n)$, where $x_i = c_{|\pi_1|} \operatorname{sgn}(\pi_{|\pi_1|}) + \cdots + c_{|\pi_i|} \operatorname{sgn}(\pi_{|\pi_i|})$ and $(c_1, \dots ,c_n) = (1, 2, \dots, 2^{n-1})$. The actual value of the c-tuple for Andersen's statistic does not matter so long as no sums or differences of any subset of the $c_i$'s is zero. The choice of powers of $2$ is just a convenient choice. This returns the largest position of the maximum value in the $x$-tuple. This is related to the ''discrete arcsine distribution''. The number of signed permutations with value equal to $k$ is given by $\binom{2k}{k} \binom{2n-2k}{n-k} \frac{n!}{2^n}$. This statistic is equidistributed with Sparre Andersen's `Number of Positives' statistic.