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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000562
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St000562: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 0
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 2
{{1,3,4},{2},{5}}
=> 2
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 0
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 1
Description
The number of internal points of a set partition.
An element $e$ is internal, if there are $f < e < g$ such that the blocks of $f$ and $g$ have larger minimal element than the block of $e$. See Section 5.5 of [1]
Matching statistic: St001687
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 83%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 83%
Values
{{1,2}}
=> [2,1] => [1,2] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 0
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 0
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => 1
{{1,4,6,7},{2,3,5}}
=> [4,3,5,6,2,7,1] => [1,4,6,7,2,3,5] => ? = 1
{{1,4,7},{2,3,5,6}}
=> [4,3,5,7,6,2,1] => [1,4,7,2,3,5,6] => ? = 1
{{1,4,7},{2,3,5},{6}}
=> [4,3,5,7,2,6,1] => [1,4,7,2,3,5,6] => ? = 1
{{1,4,6,7},{2,3},{5}}
=> [4,3,2,6,5,7,1] => [1,4,6,7,2,3,5] => ? = 1
{{1,4,7},{2,3,6},{5}}
=> [4,3,6,7,5,2,1] => [1,4,7,2,3,6,5] => ? = 1
{{1,4,7},{2,3},{5,6}}
=> [4,3,2,7,6,5,1] => [1,4,7,2,3,5,6] => ? = 1
{{1,4,7},{2,3},{5},{6}}
=> [4,3,2,7,5,6,1] => [1,4,7,2,3,5,6] => ? = 1
{{1,5,6,7},{2,3,4}}
=> [5,3,4,2,6,7,1] => [1,5,6,7,2,3,4] => ? = 0
{{1,5,6},{2,3,4,7}}
=> [5,3,4,7,6,1,2] => [1,5,6,2,3,4,7] => ? = 2
{{1,5,6},{2,3,4},{7}}
=> [5,3,4,2,6,1,7] => [1,5,6,2,3,4,7] => ? = 2
{{1,5,7},{2,3,4,6}}
=> [5,3,4,6,7,2,1] => [1,5,7,2,3,4,6] => ? = 1
{{1,5},{2,3,4,6,7}}
=> [5,3,4,6,1,7,2] => [1,5,2,3,4,6,7] => ? = 1
{{1,5},{2,3,4,6},{7}}
=> [5,3,4,6,1,2,7] => [1,5,2,3,4,6,7] => ? = 1
{{1,5,7},{2,3,4},{6}}
=> [5,3,4,2,7,6,1] => [1,5,7,2,3,4,6] => ? = 1
{{1,5},{2,3,4,7},{6}}
=> [5,3,4,7,1,6,2] => [1,5,2,3,4,7,6] => ? = 1
{{1,5},{2,3,4},{6,7}}
=> [5,3,4,2,1,7,6] => [1,5,2,3,4,6,7] => ? = 1
{{1,5},{2,3,4},{6},{7}}
=> [5,3,4,2,1,6,7] => [1,5,2,3,4,6,7] => ? = 1
{{1,6,7},{2,3,4,5}}
=> [6,3,4,5,2,7,1] => [1,6,7,2,3,4,5] => ? = 0
{{1,6},{2,3,4,5,7}}
=> [6,3,4,5,7,1,2] => [1,6,2,3,4,5,7] => ? = 1
{{1,6},{2,3,4,5},{7}}
=> [6,3,4,5,2,1,7] => [1,6,2,3,4,5,7] => ? = 1
{{1,7},{2,3,4,5,6}}
=> [7,3,4,5,6,2,1] => [1,7,2,3,4,5,6] => ? = 0
{{1,7},{2,3,4,5},{6}}
=> [7,3,4,5,2,6,1] => [1,7,2,3,4,5,6] => ? = 0
{{1,6,7},{2,3,4},{5}}
=> [6,3,4,2,5,7,1] => [1,6,7,2,3,4,5] => ? = 0
{{1,6},{2,3,4,7},{5}}
=> [6,3,4,7,5,1,2] => [1,6,2,3,4,7,5] => ? = 1
{{1,6},{2,3,4},{5,7}}
=> [6,3,4,2,7,1,5] => [1,6,2,3,4,5,7] => ? = 1
{{1,6},{2,3,4},{5},{7}}
=> [6,3,4,2,5,1,7] => [1,6,2,3,4,5,7] => ? = 1
{{1,7},{2,3,4,6},{5}}
=> [7,3,4,6,5,2,1] => [1,7,2,3,4,6,5] => ? = 0
{{1,7},{2,3,4},{5,6}}
=> [7,3,4,2,6,5,1] => [1,7,2,3,4,5,6] => ? = 0
{{1,7},{2,3,4},{5},{6}}
=> [7,3,4,2,5,6,1] => [1,7,2,3,4,5,6] => ? = 0
{{1,5,6,7},{2,3},{4}}
=> [5,3,2,4,6,7,1] => [1,5,6,7,2,3,4] => ? = 0
{{1,5,6},{2,3,7},{4}}
=> [5,3,7,4,6,1,2] => [1,5,6,2,3,7,4] => ? = 2
{{1,5,6},{2,3},{4,7}}
=> [5,3,2,7,6,1,4] => [1,5,6,2,3,4,7] => ? = 2
{{1,5,6},{2,3},{4},{7}}
=> [5,3,2,4,6,1,7] => [1,5,6,2,3,4,7] => ? = 2
{{1,5,7},{2,3,6},{4}}
=> [5,3,6,4,7,2,1] => [1,5,7,2,3,6,4] => ? = 1
{{1,5},{2,3,6,7},{4}}
=> [5,3,6,4,1,7,2] => [1,5,2,3,6,7,4] => ? = 1
{{1,5},{2,3,6},{4,7}}
=> [5,3,6,7,1,2,4] => [1,5,2,3,6,4,7] => ? = 2
{{1,5},{2,3,6},{4},{7}}
=> [5,3,6,4,1,2,7] => [1,5,2,3,6,4,7] => ? = 2
{{1,5,7},{2,3},{4,6}}
=> [5,3,2,6,7,4,1] => [1,5,7,2,3,4,6] => ? = 1
{{1,5},{2,3,7},{4,6}}
=> [5,3,7,6,1,4,2] => [1,5,2,3,7,4,6] => ? = 1
{{1,5},{2,3},{4,6,7}}
=> [5,3,2,6,1,7,4] => [1,5,2,3,4,6,7] => ? = 1
{{1,5},{2,3},{4,6},{7}}
=> [5,3,2,6,1,4,7] => [1,5,2,3,4,6,7] => ? = 1
{{1,5,7},{2,3},{4},{6}}
=> [5,3,2,4,7,6,1] => [1,5,7,2,3,4,6] => ? = 1
{{1,5},{2,3,7},{4},{6}}
=> [5,3,7,4,1,6,2] => [1,5,2,3,7,4,6] => ? = 1
{{1,5},{2,3},{4,7},{6}}
=> [5,3,2,7,1,6,4] => [1,5,2,3,4,7,6] => ? = 1
{{1,5},{2,3},{4},{6,7}}
=> [5,3,2,4,1,7,6] => [1,5,2,3,4,6,7] => ? = 1
{{1,5},{2,3},{4},{6},{7}}
=> [5,3,2,4,1,6,7] => [1,5,2,3,4,6,7] => ? = 1
{{1,6,7},{2,3,5},{4}}
=> [6,3,5,4,2,7,1] => [1,6,7,2,3,5,4] => ? = 0
{{1,6},{2,3,5,7},{4}}
=> [6,3,5,4,7,1,2] => [1,6,2,3,5,7,4] => ? = 1
{{1,6},{2,3,5},{4,7}}
=> [6,3,5,7,2,1,4] => [1,6,2,3,5,4,7] => ? = 2
{{1,6},{2,3,5},{4},{7}}
=> [6,3,5,4,2,1,7] => [1,6,2,3,5,4,7] => ? = 2
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
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