Your data matches 19 different statistics following compositions of up to 3 maps.
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Mp00081: Standard tableaux reading word permutationPermutations
Mp00069: Permutations complementPermutations
Mp00109: Permutations descent wordBinary words
St000293: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => 1 => 0
[[1],[2]]
=> [2,1] => [1,2] => 0 => 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 11 => 0
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 01 => 0
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 01 => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 00 => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 111 => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [3,4,2,1] => 011 => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => 011 => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 011 => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 101 => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 101 => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => 001 => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => 001 => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 001 => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 000 => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 1111 => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [4,5,3,2,1] => 0111 => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,5,4,2,1] => 0111 => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,5,4,3,1] => 0111 => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0111 => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,5,3,1] => 1011 => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => 1011 => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,1,5,3,2] => 1011 => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => 1011 => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 1011 => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => 0011 => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => 0011 => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [2,3,5,4,1] => 0011 => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 0011 => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 0011 => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 0011 => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,4,1,5,2] => 0101 => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 0101 => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 0101 => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,4,2,5,3] => 0101 => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 0101 => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,3,4,5,1] => 0001 => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 0001 => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 0001 => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 0001 => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 11111 => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [5,6,4,3,2,1] => 01111 => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [4,6,5,3,2,1] => 01111 => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [3,6,5,4,2,1] => 01111 => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,6,5,4,3,1] => 01111 => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 01111 => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [5,3,6,4,2,1] => 10111 => 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => 10111 => 1
Description
The number of inversions of a binary word.
Matching statistic: St000566
Mp00083: Standard tableaux shapeInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000566: Integer partitions ⟶ ℤResult quality: 76% values known / values provided: 76%distinct values known / distinct values provided: 90%
Values
[[1,2]]
=> [2]
=> [1,1]
=> [1]
=> ? = 0
[[1],[2]]
=> [1,1]
=> [2]
=> []
=> ? = 0
[[1,2,3]]
=> [3]
=> [1,1,1]
=> [1,1]
=> 0
[[1,3],[2]]
=> [2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1,2],[3]]
=> [2,1]
=> [2,1]
=> [1]
=> ? = 0
[[1],[2],[3]]
=> [1,1,1]
=> [3]
=> []
=> ? = 0
[[1,2,3,4]]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2,2]
=> [2]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? = 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? = 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [4]
=> []
=> ? = 0
[[1,2,3,4,5]]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [3,2]
=> [2]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [5]
=> []
=> ? = 0
[[1,2,3,4,5,6]]
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,3,5],[4,6]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,3,4],[5,6]]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? = 0
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? = 0
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? = 0
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? = 0
[[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> ? = 0
[[1,7],[2],[3],[4],[5],[6]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? = 0
[[1,6],[2],[3],[4],[5],[7]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? = 0
[[1,5],[2],[3],[4],[6],[7]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? = 0
[[1,4],[2],[3],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? = 0
[[1,3],[2],[4],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? = 0
[[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1]
=> [7]
=> []
=> ? = 0
[[1,8],[2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1,7],[2],[3],[4],[5],[6],[8]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1,6],[2],[3],[4],[5],[7],[8]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1,5],[2],[3],[4],[6],[7],[8]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1,4],[2],[3],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1,3],[2],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> []
=> ? = 0
[[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> []
=> ? = 0
[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1]
=> ? = 0
[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> []
=> ? = 0
[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> ?
=> ?
=> ?
=> ? = 15
[[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1]
=> ? = 0
[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1]
=> ? = 0
[[1,8],[2,9],[3],[4],[5],[6],[7]]
=> ?
=> ?
=> ?
=> ? = 1
[[1,7],[2,8],[3,9],[4],[5],[6]]
=> ?
=> ?
=> ?
=> ? = 3
[[1,7,9],[2,8],[3],[4],[5],[6]]
=> ?
=> ?
=> ?
=> ? = 1
[[1,6],[2,7],[3,8],[4,9],[5]]
=> ?
=> ?
=> ?
=> ? = 6
[[1,6,9],[2,7],[3,8],[4],[5]]
=> ?
=> ?
=> ?
=> ? = 3
[[1,6,8],[2,7,9],[3],[4],[5]]
=> ?
=> ?
=> ?
=> ? = 2
[[1,6,8,9],[2,7],[3],[4],[5]]
=> ?
=> ?
=> ?
=> ? = 1
[[1,5,9],[2,6],[3,7],[4,8]]
=> ?
=> ?
=> ?
=> ? = 6
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 $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Mp00084: Standard tableaux conjugateStandard tableaux
St000017: Standard tableaux ⟶ ℤResult quality: 60% values known / values provided: 63%distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [[1],[2]]
=> 0
[[1],[2]]
=> [[1,2]]
=> 0
[[1,2,3]]
=> [[1],[2],[3]]
=> 0
[[1,3],[2]]
=> [[1,2],[3]]
=> 0
[[1,2],[3]]
=> [[1,3],[2]]
=> 0
[[1],[2],[3]]
=> [[1,2,3]]
=> 0
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 0
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 0
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 0
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 1
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 0
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 0
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 0
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 0
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 0
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 0
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 1
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 0
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 0
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 0
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 1
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 1
[[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 1
[[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 4
[[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ? = 4
[[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10]]
=> ? = 4
[[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10]]
=> ? = 4
[[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2],[3,4],[5,8],[6,9],[7,10]]
=> ? = 4
[[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10]]
=> ? = 4
[[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ? = 4
[[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ? = 4
[[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10]]
=> ? = 4
[[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2],[3,5],[4,8],[6,9],[7,10]]
=> ? = 4
[[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ? = 4
[[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2],[3,6],[4,7],[5,9],[8,10]]
=> ? = 4
[[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2],[3,6],[4,8],[5,9],[7,10]]
=> ? = 4
[[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2],[3,7],[4,8],[5,9],[6,10]]
=> ? = 4
[[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10]]
=> ? = 4
[[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10]]
=> ? = 4
[[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,3],[2,4],[5,7],[6,8],[9,10]]
=> ? = 4
[[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10]]
=> ? = 4
[[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,3],[2,4],[5,8],[6,9],[7,10]]
=> ? = 4
[[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ? = 4
[[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ? = 4
[[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 4
[[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ? = 4
[[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,3],[2,5],[4,8],[6,9],[7,10]]
=> ? = 4
[[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 4
[[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,3],[2,6],[4,7],[5,9],[8,10]]
=> ? = 4
[[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,3],[2,6],[4,8],[5,9],[7,10]]
=> ? = 4
[[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,3],[2,7],[4,8],[5,9],[6,10]]
=> ? = 4
[[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ? = 4
[[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ? = 4
[[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 4
[[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ? = 4
[[1,2,3,6,7],[4,5,8,9,10]]
=> [[1,4],[2,5],[3,8],[6,9],[7,10]]
=> ? = 4
[[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 4
[[1,2,3,5,8],[4,6,7,9,10]]
=> [[1,4],[2,6],[3,7],[5,9],[8,10]]
=> ? = 4
[[1,2,3,5,7],[4,6,8,9,10]]
=> [[1,4],[2,6],[3,8],[5,9],[7,10]]
=> ? = 4
[[1,2,3,5,6],[4,7,8,9,10]]
=> [[1,4],[2,7],[3,8],[5,9],[6,10]]
=> ? = 4
[[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,5],[2,6],[3,7],[4,8],[9,10]]
=> ? = 4
[[1,2,3,4,8],[5,6,7,9,10]]
=> [[1,5],[2,6],[3,7],[4,9],[8,10]]
=> ? = 4
[[1,2,3,4,7],[5,6,8,9,10]]
=> [[1,5],[2,6],[3,8],[4,9],[7,10]]
=> ? = 4
[[1,2,3,4,6],[5,7,8,9,10]]
=> [[1,5],[2,7],[3,8],[4,9],[6,10]]
=> ? = 4
[[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> ? = 4
[[1,2,3,4,5,6,7,8,9]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[[1,2,3,4,5,6,7,8],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[[1,2,3,4,5,6,7],[8,9]]
=> [[1,8],[2,9],[3],[4],[5],[6],[7]]
=> ? = 1
[[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> ? = 2
[[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> ? = 1
[[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> ? = 0
[[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> ? = 3
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 $d$.
Mp00081: Standard tableaux reading word permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000609: Set partitions ⟶ ℤResult quality: 26% values known / values provided: 26%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [[1,2]]
=> {{1,2}}
=> 0
[[1],[2]]
=> [2,1] => [[1],[2]]
=> {{1},{2}}
=> 0
[[1,2,3]]
=> [1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> [2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[[1,2],[3]]
=> [3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 0
[[1],[2],[3]]
=> [3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 0
[[1,2,4],[3]]
=> [3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 0
[[1,2,3],[4]]
=> [4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 0
[[1,3],[2,4]]
=> [2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 1
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => ?
=> ?
=> ? = 1
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => ?
=> ?
=> ? = 0
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => ?
=> ?
=> ? = 0
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> [2,4,7,1,3,5,6,8] => ?
=> ?
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> [3,4,7,1,2,5,6,8] => ?
=> ?
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> [2,5,7,1,3,4,6,8] => ?
=> ?
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> [3,5,7,1,2,4,6,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> [2,6,7,1,3,4,5,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [3,6,7,1,2,4,5,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> [4,6,7,1,2,3,5,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> [5,6,7,1,2,3,4,8] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> [2,4,8,1,3,5,6,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> [3,4,8,1,2,5,6,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> [2,5,8,1,3,4,6,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> [3,5,8,1,2,4,6,7] => ?
=> ?
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [4,5,8,1,2,3,6,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [2,6,8,1,3,4,5,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> [3,6,8,1,2,4,5,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [4,6,8,1,2,3,5,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> [5,6,8,1,2,3,4,7] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> [2,7,8,1,3,4,5,6] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [3,7,8,1,2,4,5,6] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> [4,7,8,1,2,3,5,6] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> [5,7,8,1,2,3,4,6] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 2
[[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ?
=> ?
=> ? = 1
[[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,6,7,8],[3,5],[4]]
=> [4,3,5,1,2,6,7,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,3,6,7,8],[2,4],[5]]
=> [5,2,4,1,3,6,7,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,6,7,8],[3,4],[5]]
=> [5,3,4,1,2,6,7,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,3,4,7,8],[2,6],[5]]
=> [5,2,6,1,3,4,7,8] => ?
=> ?
=> ? = 1
[[1,2,4,7,8],[3,6],[5]]
=> [5,3,6,1,2,4,7,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,4,7,8],[3,5],[6]]
=> [6,3,5,1,2,4,7,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,3,5,6,8],[2,7],[4]]
=> [4,2,7,1,3,5,6,8] => ?
=> ?
=> ? = 1
[[1,2,5,6,8],[3,7],[4]]
=> [4,3,7,1,2,5,6,8] => ?
=> ?
=> ? = 1
[[1,2,4,6,8],[3,7],[5]]
=> [5,3,7,1,2,4,6,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,3,4,5,8],[2,7],[6]]
=> [6,2,7,1,3,4,5,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,4,5,8],[3,7],[6]]
=> [6,3,7,1,2,4,5,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,3,5,8],[4,7],[6]]
=> [6,4,7,1,2,3,5,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,3,4,8],[5,7],[6]]
=> [6,5,7,1,2,3,4,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,3,4,6,8],[2,5],[7]]
=> [7,2,5,1,3,4,6,8] => ?
=> ?
=> ? = 1
[[1,2,4,6,8],[3,5],[7]]
=> [7,3,5,1,2,4,6,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,3,6,8],[4,5],[7]]
=> [7,4,5,1,2,3,6,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,4,5,8],[3,6],[7]]
=> [7,3,6,1,2,4,5,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,2,3,4,8],[5,6],[7]]
=> [7,5,6,1,2,3,4,8] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,4,5,6,7],[2,8],[3]]
=> [3,2,8,1,4,5,6,7] => [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 1
[[1,3,4,6,7],[2,8],[5]]
=> [5,2,8,1,3,4,6,7] => ?
=> ?
=> ? = 1
[[1,2,4,6,7],[3,8],[5]]
=> [5,3,8,1,2,4,6,7] => ?
=> ?
=> ? = 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000497
Mp00284: Standard tableaux rowsSet partitions
Mp00219: Set partitions inverse YipSet partitions
Mp00217: Set partitions Wachs-White-rho Set partitions
St000497: Set partitions ⟶ ℤResult quality: 25% values known / values provided: 25%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> 0
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> 0
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> 0
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> 0
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> 0
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1},{2,3,4,5,6}}
=> {{1},{2,3,4,5,6}}
=> 0
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2},{3,4,5,6}}
=> {{1,2},{3,4,5,6}}
=> 0
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3},{4,5,6}}
=> {{1,2,3},{4,5,6}}
=> 0
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> {{1,2,3,4},{5,6}}
=> 0
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,4,5,6},{2,3}}
=> {{1,3},{2,4,5,6}}
=> 1
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,4,5,6},{3}}
=> {{1,2,4,5,6},{3}}
=> 1
[[1,2,3,5,6,7,8],[4]]
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,2,3},{4,5,6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[[1,2,3,4,6,7,8],[5]]
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[[1,2,3,4,5,7,8],[6]]
=> {{1,2,3,4,5,7,8},{6}}
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0
[[1,3,4,6,7,8],[2,5]]
=> {{1,3,4,6,7,8},{2,5}}
=> {{1,5,6,7,8},{2,3,4}}
=> {{1,3,4},{2,5,6,7,8}}
=> ? = 1
[[1,2,4,6,7,8],[3,5]]
=> {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5,6,7,8},{3,4}}
=> {{1,2,4},{3,5,6,7,8}}
=> ? = 1
[[1,3,4,5,7,8],[2,6]]
=> {{1,3,4,5,7,8},{2,6}}
=> {{1,6,7,8},{2,3,4,5}}
=> {{1,3,4,5},{2,6,7,8}}
=> ? = 1
[[1,2,4,5,7,8],[3,6]]
=> {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6,7,8},{3,4,5}}
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 1
[[1,2,3,5,7,8],[4,6]]
=> {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6,7,8},{4,5}}
=> {{1,2,3,5},{4,6,7,8}}
=> ? = 1
[[1,3,4,5,6,8],[2,7]]
=> {{1,3,4,5,6,8},{2,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> {{1,3,4,5,6},{2,7,8}}
=> ? = 1
[[1,2,4,5,6,8],[3,7]]
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,2,7,8},{3,4,5,6}}
=> {{1,2,4,5,6},{3,7,8}}
=> ? = 1
[[1,2,3,5,6,8],[4,7]]
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7,8},{4,5,6}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 1
[[1,2,3,4,6,8],[5,7]]
=> {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7,8},{5,6}}
=> {{1,2,3,4,6},{5,7,8}}
=> ? = 1
[[1,3,5,6,7,8],[2],[4]]
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1},{2,3},{4,5,6,7,8}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 0
[[1,2,5,6,7,8],[3],[4]]
=> {{1,2,5,6,7,8},{3},{4}}
=> {{1,2},{3},{4,5,6,7,8}}
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 0
[[1,3,4,6,7,8],[2],[5]]
=> {{1,3,4,6,7,8},{2},{5}}
=> {{1},{2,3,4},{5,6,7,8}}
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 0
[[1,2,4,6,7,8],[3],[5]]
=> {{1,2,4,6,7,8},{3},{5}}
=> {{1,2},{3,4},{5,6,7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> {{1,2,3},{4},{5,6,7,8}}
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[[1,3,4,5,7,8],[2],[6]]
=> {{1,3,4,5,7,8},{2},{6}}
=> {{1},{2,3,4,5},{6,7,8}}
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 0
[[1,2,4,5,7,8],[3],[6]]
=> {{1,2,4,5,7,8},{3},{6}}
=> {{1,2},{3,4,5},{6,7,8}}
=> {{1,2},{3,4,5},{6,7,8}}
=> ? = 0
[[1,2,3,5,7,8],[4],[6]]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,3},{4,5},{6,7,8}}
=> ?
=> ? = 0
[[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,3,4},{5},{6,7,8}}
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 0
[[1,3,4,5,6,8],[2],[7]]
=> {{1,3,4,5,6,8},{2},{7}}
=> {{1},{2,3,4,5,6},{7,8}}
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 0
[[1,2,4,5,6,8],[3],[7]]
=> {{1,2,4,5,6,8},{3},{7}}
=> {{1,2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
[[1,2,3,5,6,8],[4],[7]]
=> {{1,2,3,5,6,8},{4},{7}}
=> {{1,2,3},{4,5,6},{7,8}}
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 0
[[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> {{1,2,3,4},{5,6},{7,8}}
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[[1,2,3,4,5,8],[6],[7]]
=> {{1,2,3,4,5,8},{6},{7}}
=> {{1,2,3,4,5},{6},{7,8}}
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[[1,2,4,5,6,7],[3],[8]]
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1,2},{3,4,5,6,7},{8}}
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 0
[[1,2,3,5,6,7],[4],[8]]
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1,2,3},{4,5,6,7},{8}}
=> {{1,2,3},{4,5,6,7},{8}}
=> ? = 0
[[1,2,3,4,6,7],[5],[8]]
=> {{1,2,3,4,6,7},{5},{8}}
=> {{1,2,3,4},{5,6,7},{8}}
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[[1,2,3,4,5,7],[6],[8]]
=> {{1,2,3,4,5,7},{6},{8}}
=> {{1,2,3,4,5},{6,7},{8}}
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 0
[[1,2,3,4,5,6],[7],[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}}
=> ? = 0
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5},{2,3,6,7,8}}
=> {{1,3,5},{2,4,6,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> {{1,3,5,6},{2,4,7,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,5,6},{3,4,7,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> {{1,5,6},{2,3,4,7,8}}
=> {{1,3,4,6},{2,5,7,8}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> {{1,4,5,6,7},{2,3,8}}
=> {{1,3,5,6,7},{2,4,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> {{1,5,6,7},{2,3,4,8}}
=> {{1,3,4,6,7},{2,5,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,6,7},{3,5,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,3,5,6,7},{4,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> {{1,3,4,5,7},{2,6,8}}
=> {{1,6,7},{2,3,4,5,8}}
=> {{1,3,4,5,7},{2,6,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> {{1,2,4,5,7},{3,6,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,3,5,7},{4,6,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> {{1,2,3,4,7},{5,6,8}}
=> {{1,2,3,4,6,7},{5,8}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> {{1,3,4,5,6},{2,7,8}}
=> ?
=> ?
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,7},{3,4,5,6,8}}
=> ?
=> ? = 2
[[1,4,6,7,8],[2,5],[3]]
=> {{1,4,6,7,8},{2,5},{3}}
=> {{1},{2,5,6,7,8},{3,4}}
=> {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[[1,2,6,7,8],[3,5],[4]]
=> {{1,2,6,7,8},{3,5},{4}}
=> {{1,2},{3,5,6,7,8},{4}}
=> {{1,2},{3,5,6,7,8},{4}}
=> ? = 1
[[1,3,6,7,8],[2,4],[5]]
=> {{1,3,6,7,8},{2,4},{5}}
=> {{1,4},{2,3},{5,6,7,8}}
=> {{1,3},{2,4},{5,6,7,8}}
=> ? = 1
Description
The lcb statistic of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000589
Mp00083: Standard tableaux shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000589: Set partitions ⟶ ℤResult quality: 25% values known / values provided: 25%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[[1,2,3]]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,4,5,6,7,8],[2],[3]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,5,6,7,8],[2],[4]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,5,6,7,8],[3],[4]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,6,7,8],[2],[5]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,6,7,8],[3],[5]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,6,7,8],[4],[5]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,5,7,8],[2],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,5,7,8],[3],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,5,7,8],[4],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,7,8],[5],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,5,6,8],[2],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,5,6,8],[3],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,5,6,8],[4],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,6,8],[5],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,5,8],[6],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,5,6,7],[2],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,5,6,7],[3],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,5,6,7],[4],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,6,7],[5],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,5,7],[6],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,5,6],[7],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,5,7,8],[2,4,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,4,6,7,8],[2,5],[3]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,3,6,7,8],[2,5],[4]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,2,6,7,8],[3,5],[4]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,3,6,7,8],[2,4],[5]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,2,6,7,8],[3,4],[5]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000612
Mp00083: Standard tableaux shapeInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St000612: Set partitions ⟶ ℤResult quality: 25% values known / values provided: 25%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> [[1,2]]
=> {{1,2}}
=> 0
[[1],[2]]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0
[[1,2,3]]
=> [3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1,2],[3]]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> [4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> [5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,2,5,6],[3,4]]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[[1,4,5,6,7,8],[2],[3]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,5,6,7,8],[2],[4]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,5,6,7,8],[3],[4]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,6,7,8],[2],[5]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,6,7,8],[3],[5]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,6,7,8],[4],[5]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,5,7,8],[2],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,5,7,8],[3],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,5,7,8],[4],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,7,8],[5],[6]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,5,6,8],[2],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,5,6,8],[3],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,5,6,8],[4],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,6,8],[5],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,5,8],[6],[7]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,4,5,6,7],[2],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,4,5,6,7],[3],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,5,6,7],[4],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,6,7],[5],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,5,7],[6],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,2,3,4,5,6],[7],[8]]
=> [6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,5,7,8],[2,4,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,7],[4,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,7],[5,6,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,3,4,5,6],[2,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,4,5,6],[3,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,5,6],[4,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,6],[5,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,2,3,4,5],[6,7,8]]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 2
[[1,4,6,7,8],[2,5],[3]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,3,6,7,8],[2,5],[4]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,2,6,7,8],[3,5],[4]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,3,6,7,8],[2,4],[5]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
[[1,2,6,7,8],[3,4],[5]]
=> [5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 1
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000491
Mp00284: Standard tableaux rowsSet partitions
Mp00219: Set partitions inverse YipSet partitions
Mp00215: Set partitions Wachs-WhiteSet partitions
St000491: Set partitions ⟶ ℤResult quality: 24% values known / values provided: 24%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 0
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 0
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4,5},{3}}
=> {{1,2,4},{3,5}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,4},{2,3,5}}
=> 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,3},{2,4,5}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 0
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 0
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 0
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 0
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1},{2,3,4,5,6}}
=> {{1,2,3,4,5},{6}}
=> 0
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2},{3,4,5,6}}
=> {{1,2,3,4},{5,6}}
=> 0
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3},{4,5,6}}
=> {{1,2,3},{4,5,6}}
=> 0
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> {{1,2},{3,4,5,6}}
=> 0
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,4,5,6},{2,3}}
=> {{1,2,3,6},{4,5}}
=> 1
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5},{4,6}}
=> 1
[[1,2,3,5,6,7,8],[4]]
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,2,3},{4,5,6,7,8}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 0
[[1,2,3,4,6,7,8],[5]]
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 0
[[1,2,3,4,5,7,8],[6]]
=> {{1,2,3,4,5,7,8},{6}}
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0
[[1,3,4,6,7,8],[2,5]]
=> {{1,3,4,6,7,8},{2,5}}
=> {{1,5,6,7,8},{2,3,4}}
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 1
[[1,2,4,6,7,8],[3,5]]
=> {{1,2,4,6,7,8},{3,5}}
=> {{1,2,5,6,7,8},{3,4}}
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 1
[[1,3,4,5,7,8],[2,6]]
=> {{1,3,4,5,7,8},{2,6}}
=> {{1,6,7,8},{2,3,4,5}}
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 1
[[1,2,4,5,7,8],[3,6]]
=> {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6,7,8},{3,4,5}}
=> {{1,2,3,7},{4,5,6,8}}
=> ? = 1
[[1,2,3,5,7,8],[4,6]]
=> {{1,2,3,5,7,8},{4,6}}
=> {{1,2,3,6,7,8},{4,5}}
=> {{1,2,3,6},{4,5,7,8}}
=> ? = 1
[[1,2,3,4,7,8],[5,6]]
=> {{1,2,3,4,7,8},{5,6}}
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,2,3,5},{4,6,7,8}}
=> ? = 1
[[1,3,4,5,6,8],[2,7]]
=> {{1,3,4,5,6,8},{2,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 1
[[1,2,4,5,6,8],[3,7]]
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,2,7,8},{3,4,5,6}}
=> {{1,2,7},{3,4,5,6,8}}
=> ? = 1
[[1,2,3,5,6,8],[4,7]]
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7,8},{4,5,6}}
=> {{1,2,6},{3,4,5,7,8}}
=> ? = 1
[[1,2,3,4,6,8],[5,7]]
=> {{1,2,3,4,6,8},{5,7}}
=> {{1,2,3,4,7,8},{5,6}}
=> {{1,2,5},{3,4,6,7,8}}
=> ? = 1
[[1,2,3,4,5,8],[6,7]]
=> {{1,2,3,4,5,8},{6,7}}
=> {{1,2,3,4,5,7,8},{6}}
=> {{1,2,4},{3,5,6,7,8}}
=> ? = 1
[[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> {{1},{2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 0
[[1,3,5,6,7,8],[2],[4]]
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1},{2,3},{4,5,6,7,8}}
=> {{1,2,3,4,5},{6,7},{8}}
=> ? = 0
[[1,2,5,6,7,8],[3],[4]]
=> {{1,2,5,6,7,8},{3},{4}}
=> {{1,2},{3},{4,5,6,7,8}}
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 0
[[1,3,4,6,7,8],[2],[5]]
=> {{1,3,4,6,7,8},{2},{5}}
=> {{1},{2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 0
[[1,2,4,6,7,8],[3],[5]]
=> {{1,2,4,6,7,8},{3},{5}}
=> {{1,2},{3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 0
[[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> {{1,2,3},{4},{5,6,7,8}}
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 0
[[1,3,4,5,7,8],[2],[6]]
=> {{1,3,4,5,7,8},{2},{6}}
=> {{1},{2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7},{8}}
=> ? = 0
[[1,2,4,5,7,8],[3],[6]]
=> {{1,2,4,5,7,8},{3},{6}}
=> {{1,2},{3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 0
[[1,2,3,5,7,8],[4],[6]]
=> {{1,2,3,5,7,8},{4},{6}}
=> {{1,2,3},{4,5},{6,7,8}}
=> ?
=> ? = 0
[[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> {{1,2,3,4},{5},{6,7,8}}
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 0
[[1,3,4,5,6,8],[2],[7]]
=> {{1,3,4,5,6,8},{2},{7}}
=> {{1},{2,3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6,7},{8}}
=> ? = 0
[[1,2,4,5,6,8],[3],[7]]
=> {{1,2,4,5,6,8},{3},{7}}
=> {{1,2},{3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 0
[[1,2,3,5,6,8],[4],[7]]
=> {{1,2,3,5,6,8},{4},{7}}
=> {{1,2,3},{4,5,6},{7,8}}
=> {{1,2},{3,4,5},{6,7,8}}
=> ? = 0
[[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> {{1,2,3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 0
[[1,2,3,4,5,8],[6],[7]]
=> {{1,2,3,4,5,8},{6},{7}}
=> {{1,2,3,4,5},{6},{7,8}}
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 0
[[1,2,4,5,6,7],[3],[8]]
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1,2},{3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 0
[[1,2,3,5,6,7],[4],[8]]
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1,2,3},{4,5,6,7},{8}}
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 0
[[1,2,3,4,6,7],[5],[8]]
=> {{1,2,3,4,6,7},{5},{8}}
=> {{1,2,3,4},{5,6,7},{8}}
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 0
[[1,2,3,4,5,7],[6],[8]]
=> {{1,2,3,4,5,7},{6},{8}}
=> {{1,2,3,4,5},{6,7},{8}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 0
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> {{1,4,5},{2,3,6,7,8}}
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 2
[[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> {{1,2,3,5},{4,6,7,8}}
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 2
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> {{1,4,5,6},{2,3,7,8}}
=> {{1,2,6,8},{3,4,5,7}}
=> ? = 2
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> {{1,2,4,5,6},{3,7,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> ? = 2
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> {{1,5,6},{2,3,4,7,8}}
=> {{1,2,5,8},{3,4,6,7}}
=> ? = 2
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> {{1,2,5,6},{3,4,7,8}}
=> {{1,2,5,7},{3,4,6,8}}
=> ? = 2
[[1,3,4,5,8],[2,6,7]]
=> {{1,3,4,5,8},{2,6,7}}
=> {{1,6},{2,3,4,5,7,8}}
=> {{1,2,4,8},{3,5,6,7}}
=> ? = 2
[[1,2,4,5,8],[3,6,7]]
=> {{1,2,4,5,8},{3,6,7}}
=> {{1,2,6},{3,4,5,7,8}}
=> {{1,2,4,7},{3,5,6,8}}
=> ? = 2
[[1,2,3,5,8],[4,6,7]]
=> {{1,2,3,5,8},{4,6,7}}
=> {{1,2,3,6},{4,5,7,8}}
=> {{1,2,4,6},{3,5,7,8}}
=> ? = 2
[[1,2,3,4,8],[5,6,7]]
=> {{1,2,3,4,8},{5,6,7}}
=> {{1,2,3,4,6},{5,7,8}}
=> {{1,2,4,5},{3,6,7,8}}
=> ? = 2
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> {{1,4,5,6,7},{2,3,8}}
=> {{1,6,8},{2,3,4,5,7}}
=> ? = 2
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,6,7},{2,3,4,5,8}}
=> ? = 2
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> {{1,5,6,7},{2,3,4,8}}
=> {{1,5,8},{2,3,4,6,7}}
=> ? = 2
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> {{1,2,5,6,7},{3,4,8}}
=> {{1,5,7},{2,3,4,6,8}}
=> ? = 2
[[1,2,3,6,7],[4,5,8]]
=> {{1,2,3,6,7},{4,5,8}}
=> {{1,2,3,5,6,7},{4,8}}
=> {{1,5,6},{2,3,4,7,8}}
=> ? = 2
[[1,3,4,5,7],[2,6,8]]
=> {{1,3,4,5,7},{2,6,8}}
=> {{1,6,7},{2,3,4,5,8}}
=> {{1,4,8},{2,3,5,6,7}}
=> ? = 2
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> {{1,2,6,7},{3,4,5,8}}
=> {{1,4,7},{2,3,5,6,8}}
=> ? = 2
Description
The number of inversions of a set partition. Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$. According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$. This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller". This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Mp00081: Standard tableaux reading word permutationPermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00069: Permutations complementPermutations
St000516: Permutations ⟶ ℤResult quality: 12% values known / values provided: 12%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [1,2] => [2,1] => 0
[[1],[2]]
=> [2,1] => [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [1,3,2] => [3,1,2] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,3,1] => 0
[[1,2],[3]]
=> [3,1,2] => [3,1,2] => [1,3,2] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,5,4,2] => [3,5,1,2,4] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1,5,3,2] => [2,5,1,3,4] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,4,3,2] => [1,5,2,3,4] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,5,1,4,2] => [3,1,5,2,4] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,5,1,4,3] => [4,1,5,2,3] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,5,1,4,2] => [3,1,5,2,4] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [4,5,1,3,2] => [2,1,5,3,4] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,5,4] => [3,4,5,1,2] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [4,2,1,5,3] => [2,4,5,1,3] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,3,1,5,2] => [2,3,5,1,4] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [5,2,1,4,3] => [1,4,5,2,3] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,2,4] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [5,4,1,3,2] => [1,2,5,3,4] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,5,1,4] => [3,4,1,5,2] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,5,1,2] => [2,3,1,5,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,3,4,1,2] => [1,3,2,5,4] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,3,4,5,1] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,6,5,4,3,2] => [6,1,2,3,4,5] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,6,5,4,3] => [5,6,1,2,3,4] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => [4,6,1,2,3,5] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,6,5,3,2] => [3,6,1,2,4,5] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1,6,4,3,2] => [2,6,1,3,4,5] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,6,2,3,4,5] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,6,1,5,4,3] => [5,1,6,2,3,4] => 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,6,1,5,4,2] => [4,1,6,2,3,5] => 1
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [7,1,2,3,4,5,6] => ? = 0
[[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => ? = 0
[[1,2,4,5,6,7],[3]]
=> [3,1,2,4,5,6,7] => [3,1,7,6,5,4,2] => [5,7,1,2,3,4,6] => ? = 0
[[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [4,1,7,6,5,3,2] => [4,7,1,2,3,5,6] => ? = 0
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [5,1,7,6,4,3,2] => [3,7,1,2,4,5,6] => ? = 0
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [6,1,7,5,4,3,2] => [2,7,1,3,4,5,6] => ? = 0
[[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 0
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [4,7,1,6,5,3,2] => [4,1,7,2,3,5,6] => ? = 1
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,7,1,6,5,3,2] => [4,1,7,2,3,5,6] => ? = 1
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [5,7,1,6,4,3,2] => [3,1,7,2,4,5,6] => ? = 1
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [3,7,1,6,5,4,2] => [5,1,7,2,3,4,6] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [4,7,1,6,5,3,2] => [4,1,7,2,3,5,6] => ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [5,7,1,6,4,3,2] => [3,1,7,2,4,5,6] => ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [6,7,1,5,4,3,2] => [2,1,7,3,4,5,6] => ? = 1
[[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => [5,6,7,1,2,3,4] => ? = 0
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [4,2,1,7,6,5,3] => [4,6,7,1,2,3,5] => ? = 0
[[1,2,5,6,7],[3],[4]]
=> [4,3,1,2,5,6,7] => [4,3,1,7,6,5,2] => [4,5,7,1,2,3,6] => ? = 0
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [5,2,1,7,6,4,3] => [3,6,7,1,2,4,5] => ? = 0
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [5,3,1,7,6,4,2] => [3,5,7,1,2,4,6] => ? = 0
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [5,4,1,7,6,3,2] => [3,4,7,1,2,5,6] => ? = 0
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [6,2,1,7,5,4,3] => [2,6,7,1,3,4,5] => ? = 0
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [2,5,7,1,3,4,6] => ? = 0
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [6,4,1,7,5,3,2] => [2,4,7,1,3,5,6] => ? = 0
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [6,5,1,7,4,3,2] => [2,3,7,1,4,5,6] => ? = 0
[[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => [7,2,1,6,5,4,3] => [1,6,7,2,3,4,5] => ? = 0
[[1,2,4,5,6],[3],[7]]
=> [7,3,1,2,4,5,6] => [7,3,1,6,5,4,2] => [1,5,7,2,3,4,6] => ? = 0
[[1,2,3,5,6],[4],[7]]
=> [7,4,1,2,3,5,6] => [7,4,1,6,5,3,2] => [1,4,7,2,3,5,6] => ? = 0
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,7,6,1,5,3,2] => [4,1,2,7,3,5,6] => ? = 2
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [4,7,6,1,5,3,2] => [4,1,2,7,3,5,6] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [3,7,6,1,5,4,2] => [5,1,2,7,3,4,6] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [4,7,6,1,5,3,2] => [4,1,2,7,3,5,6] => ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [5,7,6,1,4,3,2] => [3,1,2,7,4,5,6] => ? = 2
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [3,2,7,1,6,5,4] => [5,6,1,7,2,3,4] => ? = 1
[[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [4,2,7,1,6,5,3] => [4,6,1,7,2,3,5] => ? = 1
Description
The number of stretching pairs of a permutation. This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000355
Mp00081: Standard tableaux reading word permutationPermutations
Mp00061: Permutations to increasing treeBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
St000355: Permutations ⟶ ℤResult quality: 8% values known / values provided: 8%distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [6,5,4,2,1,3] => 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [6,5,4,2,1,3] => 1
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 0
[[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => ? = 0
[[1,2,4,5,6,7],[3]]
=> [3,1,2,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => ? = 0
[[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => ? = 0
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => ? = 0
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => ? = 0
[[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => ? = 0
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1
[[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,5,6,7],[3],[4]]
=> [4,3,1,2,5,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,4,5,6],[3],[7]]
=> [7,3,1,2,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,3,5,6],[4],[7]]
=> [7,4,1,2,3,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => ? = 0
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 2
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001960The number of descents of a permutation minus one if its first entry is not one. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001427The number of descents of a signed permutation. St001487The number of inner corners of a skew partition. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.