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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 0
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,1,6,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [6,3,2,1,4,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [4,2,1,6,3,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,1,6,3,2,5] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,2,1,3,6,4] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,6,5,2,1,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,6,2,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [6,4,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,5,1,6,4,3] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 4
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,1,6,4,2] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [4,3,6,2,1,5] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [4,3,1,6,5,2] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,2,1,5,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,1,6,5,4,2] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [7,2,1,4,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [3,1,7,4,2,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> 4
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [7,3,5,2,1,4,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [7,2,4,1,5,3,6] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 2
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [4,7,3,1,5,2,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [5,7,2,1,4,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[[1,2,6],[3,4,7],[5]]
=> [5,3,4,7,1,2,6] => [3,5,1,7,4,2,6] => ([(0,5),(1,2),(1,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 4
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [4,6,2,1,7,3,5] => ([(0,1),(0,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 3
[[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [4,6,1,7,3,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [5,6,2,1,3,7,4] => ([(0,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [3,5,1,6,2,7,4] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4
[[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001488
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[[1]]
=> [[1]]
=> [1] => [[1],[]]
=> 1 = 0 + 1
[[1],[2]]
=> [[1],[2]]
=> [2] => [[2],[]]
=> 2 = 1 + 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [3] => [[3],[]]
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [3,1] => [[3,3],[2]]
=> 3 = 2 + 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [2,2] => [[3,2],[1]]
=> 4 = 3 + 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [4] => [[4],[]]
=> 2 = 1 + 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [3,2] => [[4,3],[2]]
=> 4 = 3 + 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [[3,3,2],[2,1]]
=> 5 = 4 + 1
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [2,3] => [[4,2],[1]]
=> 4 = 3 + 1
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [4,1] => [[4,4],[3]]
=> 3 = 2 + 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [3,2] => [[4,3],[2]]
=> 4 = 3 + 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [5] => [[5],[]]
=> 2 = 1 + 1
[[1,3,5],[2,4,6]]
=> [[1,3,5],[2,4,6]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 2 + 1
[[1,2,5],[3,4,6]]
=> [[1,3,4],[2,5,6]]
=> [4,2] => [[5,4],[3]]
=> ? = 2 + 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ? = 3 + 1
[[1,2,4],[3,5,6]]
=> [[1,2,4],[3,5,6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ? = 3 + 1
[[1,2,3],[4,5,6]]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [[5,3],[2]]
=> ? = 3 + 1
[[1,4,5],[2,6],[3]]
=> [[1,3,4],[2,5],[6]]
=> [4,2] => [[5,4],[3]]
=> ? = 2 + 1
[[1,3,5],[2,6],[4]]
=> [[1,3,5],[2,4],[6]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 4 + 1
[[1,2,5],[3,6],[4]]
=> [[1,3,6],[2,4],[5]]
=> [3,3] => [[5,3],[2]]
=> ? = 4 + 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ? = 3 + 1
[[1,2,4],[3,6],[5]]
=> [[1,2,6],[3,4],[5]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ? = 5 + 1
[[1,4],[2,5],[3,6]]
=> [[1,4],[2,5],[3,6]]
=> [4,2] => [[5,4],[3]]
=> ? = 3 + 1
[[1,3],[2,5],[4,6]]
=> [[1,3],[2,5],[4,6]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 2 + 1
[[1,2],[3,5],[4,6]]
=> [[1,3],[2,4],[5,6]]
=> [3,3] => [[5,3],[2]]
=> ? = 2 + 1
[[1,3],[2,4],[5,6]]
=> [[1,2],[3,5],[4,6]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ? = 4 + 1
[[1,2],[3,4],[5,6]]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ? = 3 + 1
[[1,5],[2,6],[3],[4]]
=> [[1,3],[2,4],[5],[6]]
=> [3,3] => [[5,3],[2]]
=> ? = 3 + 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,2] => [[4,3,2],[2,1]]
=> ? = 4 + 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [2,3,1] => [[4,4,2],[3,1]]
=> ? = 4 + 1
[[1,2],[3,6],[4],[5]]
=> [[1,2],[3,6],[4],[5]]
=> [2,4] => [[5,2],[1]]
=> ? = 3 + 1
[[1,4],[2,5],[3],[6]]
=> [[1,4],[2,5],[3],[6]]
=> [4,2] => [[5,4],[3]]
=> ? = 3 + 1
[[1,3],[2,5],[4],[6]]
=> [[1,3],[2,5],[4],[6]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 4 + 1
[[1,2],[3,5],[4],[6]]
=> [[1,3],[2,6],[4],[5]]
=> [3,3] => [[5,3],[2]]
=> ? = 3 + 1
[[1,3],[2,4],[5],[6]]
=> [[1,5],[2,6],[3],[4]]
=> [5,1] => [[5,5],[4]]
=> ? = 2 + 1
[[1,2],[3,4],[5],[6]]
=> [[1,4],[2,6],[3],[5]]
=> [4,2] => [[5,4],[3]]
=> ? = 3 + 1
[[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6] => [[6],[]]
=> ? = 1 + 1
[[1,3,5,6],[2,4,7]]
=> [[1,3,4,6],[2,5,7]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ? = 2 + 1
[[1,2,5,6],[3,4,7]]
=> [[1,3,4,5],[2,6,7]]
=> [5,2] => [[6,5],[4]]
=> ? = 4 + 1
[[1,4,6],[2,5,7],[3]]
=> [[1,3,5],[2,4,6],[7]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ? = 2 + 1
[[1,3,6],[2,5,7],[4]]
=> [[1,3,4],[2,5,6],[7]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ? = 2 + 1
[[1,2,6],[3,5,7],[4]]
=> [[1,3,4],[2,5,7],[6]]
=> [4,3] => [[6,4],[3]]
=> ? = 3 + 1
[[1,3,6],[2,4,7],[5]]
=> [[1,3,6],[2,4,7],[5]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ? = 3 + 1
[[1,2,6],[3,4,7],[5]]
=> [[1,3,5],[2,4,7],[6]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ? = 4 + 1
[[1,4,5],[2,6,7],[3]]
=> [[1,2,5],[3,4,6],[7]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ? = 2 + 1
[[1,3,5],[2,6,7],[4]]
=> [[1,2,4],[3,5,6],[7]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ? = 3 + 1
[[1,2,5],[3,6,7],[4]]
=> [[1,2,4],[3,5,7],[6]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ? = 3 + 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ? = 3 + 1
[[1,2,4],[3,6,7],[5]]
=> [[1,2,3],[4,5,7],[6]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ? = 4 + 1
[[1,2,3],[4,6,7],[5]]
=> [[1,2,3],[4,6,7],[5]]
=> [3,4] => [[6,3],[2]]
=> ? = 3 + 1
[[1,3,5],[2,4,7],[6]]
=> [[1,2,6],[3,4,7],[5]]
=> [2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ? = 4 + 1
[[1,2,5],[3,4,7],[6]]
=> [[1,2,5],[3,4,7],[6]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ? = 5 + 1
[[1,3,4],[2,5,7],[6]]
=> [[1,2,6],[3,5,7],[4]]
=> [2,4,1] => [[5,5,2],[4,1]]
=> ? = 3 + 1
[[1,2,4],[3,5,7],[6]]
=> [[1,2,5],[3,6,7],[4]]
=> [2,3,2] => [[5,4,2],[3,1]]
=> ? = 3 + 1
[[1,2,3],[4,5,7],[6]]
=> [[1,2,4],[3,6,7],[5]]
=> [2,2,3] => [[5,3,2],[2,1]]
=> ? = 3 + 1
[[1,3,5],[2,4,6],[7]]
=> [[1,4,6],[2,5,7],[3]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ? = 2 + 1
[[1,2,5],[3,4,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [5,2] => [[6,5],[4]]
=> ? = 2 + 1
[[1,3,4],[2,5,6],[7]]
=> [[1,3,6],[2,5,7],[4]]
=> [3,3,1] => [[5,5,3],[4,2]]
=> ? = 2 + 1
[[1,2,4],[3,5,6],[7]]
=> [[1,3,5],[2,6,7],[4]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ? = 4 + 1
[[1,2,3],[4,5,6],[7]]
=> [[1,3,4],[2,6,7],[5]]
=> [4,3] => [[6,4],[3]]
=> ? = 3 + 1
[[1,4,6],[2,5],[3,7]]
=> [[1,3,5],[2,6],[4,7]]
=> [3,2,2] => [[5,4,3],[3,2]]
=> ? = 3 + 1
[[1,3,6],[2,5],[4,7]]
=> [[1,3,4],[2,6],[5,7]]
=> [4,2,1] => [[5,5,4],[4,3]]
=> ? = 5 + 1
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
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