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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000260
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: 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
St000260: 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)
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[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)
=> 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[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)
=> 2
[[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)
=> 1
[[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)
=> 2
[[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)
=> 1
[[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)
=> 1
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 1
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 3
[[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)
=> 2
[[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)
=> 1
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 1
[[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)
=> 2
[[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)
=> 1
[[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)
=> 2
[[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)
=> 1
[[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)
=> 1
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
[[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)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000455
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(load all 5 compositions to match this statistic)
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> ? = 0 - 2
[[1],[2]]
=> [1,1] => ([(0,1)],2)
=> -1 = 1 - 2
[[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[[1,3],[2,4]]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 2
[[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[[1,4],[2,5],[3]]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1,2],[3,5],[4]]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[1,3],[2,4],[5]]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 2
[[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 1 - 2
[[1,3,5],[2,4,6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 2
[[1,2,5],[3,4,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 2
[[1,3,4],[2,5,6]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,4],[3,5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[1,4,5],[2,6],[3]]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 2
[[1,3,5],[2,6],[4]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,5],[3,6],[4]]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3,4],[2,6],[5]]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2,4],[3,6],[5]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 2
[[1,4],[2,5],[3,6]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3],[2,5],[4,6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2],[3,5],[4,6]]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 2
[[1,3],[2,4],[5,6]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,5],[2,6],[3],[4]]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3],[2,6],[4],[5]]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2],[3,6],[4],[5]]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[1,4],[2,5],[3],[6]]
=> [1,1,2,1,1] => ([(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)
=> ? = 2 - 2
[[1,3],[2,5],[4],[6]]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,2],[3,5],[4],[6]]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1,3],[2,4],[5],[6]]
=> [1,2,1,1,1] => ([(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 - 2
[[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(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 - 2
[[1,3,5,6],[2,4,7]]
=> [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 2
[[1,2,5,6],[3,4,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,4,6],[2,5,7],[3]]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 2
[[1,3,6],[2,5,7],[4]]
=> [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 2
[[1,2,6],[3,5,7],[4]]
=> [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,6],[2,4,7],[5]]
=> [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,6],[3,4,7],[5]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,4,5],[2,6,7],[3]]
=> [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,5],[2,6,7],[4]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,5],[3,6,7],[4]]
=> [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,4],[3,6,7],[5]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3],[4,6,7],[5]]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[1,3,5],[2,4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,5],[3,4,7],[6]]
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 2
[[1,3,4],[2,5,7],[6]]
=> [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,4],[3,5,7],[6]]
=> [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3],[4,5,7],[6]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,3,5],[2,4,6],[7]]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 2
[[1,2,5],[3,4,6],[7]]
=> [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 2
[[1,3,4],[2,5,6],[7]]
=> [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 2
[[1,2,4],[3,5,6],[7]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2,3],[4,5,6],[7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[[1,2],[3,7],[4],[5],[6]]
=> [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 2 - 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 1 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001520
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [[1]]
=> [1] => [1] => ? = 0 - 1
[[1],[2]]
=> [[1],[2]]
=> [2,1] => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,1,4,3] => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,5,1,4,2] => 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [3,4,1,5,2] => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,3,1,5,4] => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,2,1,6,5,3] => ? = 1 - 1
[[1,2,5],[3,4,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,4,3] => ? = 1 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [4,3,1,6,5,2] => ? = 2 - 1
[[1,2,4],[3,5,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [5,3,1,6,4,2] => ? = 2 - 1
[[1,2,3],[4,5,6]]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [5,4,1,6,3,2] => ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> [[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,6,1,5,4,3] => ? = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,6,5,1,4,3] => ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> [[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [6,2,5,1,4,3] => ? = 2 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [3,6,5,1,4,2] => ? = 2 - 1
[[1,2,4],[3,6],[5]]
=> [[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [6,3,5,1,4,2] => ? = 3 - 1
[[1,4],[2,5],[3,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,2,6,1,5,4] => ? = 2 - 1
[[1,3],[2,5],[4,6]]
=> [[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [4,2,6,1,5,3] => ? = 2 - 1
[[1,2],[3,5],[4,6]]
=> [[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,2,6,1,4,3] => ? = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [4,3,6,1,5,2] => ? = 2 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,6,1,4,2] => ? = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [2,5,6,1,4,3] => ? = 2 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [3,5,6,1,4,2] => ? = 2 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [3,4,6,1,5,2] => ? = 2 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [3,4,5,1,6,2] => ? = 2 - 1
[[1,4],[2,5],[3],[6]]
=> [[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [2,3,6,1,5,4] => ? = 2 - 1
[[1,3],[2,5],[4],[6]]
=> [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [2,4,6,1,5,3] => ? = 2 - 1
[[1,2],[3,5],[4],[6]]
=> [[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,4,5,1,6,3] => ? = 2 - 1
[[1,3],[2,4],[5],[6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [2,3,4,1,6,5] => ? = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [2,3,5,1,6,4] => ? = 2 - 1
[[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,3,5,6],[2,4,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [5,2,1,7,6,4,3] => ? = 1 - 1
[[1,2,5,6],[3,4,7]]
=> [[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [6,2,1,7,5,4,3] => ? = 2 - 1
[[1,4,6],[2,5,7],[3]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [2,7,4,1,6,5,3] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [[1,3,4],[2,5,6],[7]]
=> [7,2,5,6,1,3,4] => [2,7,5,1,6,4,3] => ? = 1 - 1
[[1,2,6],[3,5,7],[4]]
=> [[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,6,5,1,7,4,3] => ? = 2 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [2,5,4,1,7,6,3] => ? = 2 - 1
[[1,2,6],[3,4,7],[5]]
=> [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,6,4,1,7,5,3] => ? = 2 - 1
[[1,4,5],[2,6,7],[3]]
=> [[1,2,5],[3,4,6],[7]]
=> [7,3,4,6,1,2,5] => [3,7,4,1,6,5,2] => ? = 2 - 1
[[1,3,5],[2,6,7],[4]]
=> [[1,2,4],[3,5,6],[7]]
=> [7,3,5,6,1,2,4] => [3,7,5,1,6,4,2] => ? = 2 - 1
[[1,2,5],[3,6,7],[4]]
=> [[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [3,6,5,1,7,4,2] => ? = 2 - 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [4,7,5,1,6,3,2] => ? = 2 - 1
[[1,2,4],[3,6,7],[5]]
=> [[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [4,6,5,1,7,3,2] => ? = 2 - 1
[[1,2,3],[4,6,7],[5]]
=> [[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [6,4,5,1,7,3,2] => ? = 2 - 1
[[1,3,5],[2,4,7],[6]]
=> [[1,2,6],[3,4,7],[5]]
=> [5,3,4,7,1,2,6] => [3,5,4,1,7,6,2] => ? = 2 - 1
[[1,2,5],[3,4,7],[6]]
=> [[1,2,5],[3,4,7],[6]]
=> [6,3,4,7,1,2,5] => [3,6,4,1,7,5,2] => ? = 3 - 1
[[1,3,4],[2,5,7],[6]]
=> [[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [5,3,4,1,7,6,2] => ? = 2 - 1
[[1,2,4],[3,5,7],[6]]
=> [[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [6,3,4,1,7,5,2] => ? = 2 - 1
[[1,2,3],[4,5,7],[6]]
=> [[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [6,3,5,1,7,4,2] => ? = 2 - 1
[[1,3,5],[2,4,6],[7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [5,2,3,1,7,6,4] => ? = 1 - 1
[[1,2,5],[3,4,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [6,2,3,1,7,5,4] => ? = 1 - 1
[[1,3,4],[2,5,6],[7]]
=> [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [5,2,4,1,7,6,3] => ? = 1 - 1
[[1,2,4],[3,5,6],[7]]
=> [[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [6,2,4,1,7,5,3] => ? = 2 - 1
[[1,2,3],[4,5,6],[7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [6,2,5,1,7,4,3] => ? = 2 - 1
[[1,4,6],[2,5],[3,7]]
=> [[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [4,2,7,1,6,5,3] => ? = 2 - 1
Description
The number of strict 3-descents.
A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Matching statistic: St001556
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [[1]]
=> [1] => [1] => ? = 0 - 1
[[1],[2]]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0 = 1 - 1
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [3,1,2,4] => 0 = 1 - 1
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,1,2] => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [5,3,2,4,1] => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5,3,2,1,4] => 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [3,1,4,5,2] => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,1,2,5,4] => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,4,2,5] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,1,2,3] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [4,1,3,5,2,6] => ? = 1 - 1
[[1,2,5],[3,4,6]]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [4,1,6,3,5,2] => ? = 1 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [4,1,3,2,5,6] => ? = 2 - 1
[[1,2,4],[3,5,6]]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [4,1,6,3,2,5] => ? = 2 - 1
[[1,2,3],[4,5,6]]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,6,5,3,2] => ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> [[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [6,3,2,5,1,4] => ? = 1 - 1
[[1,3,5],[2,6],[4]]
=> [[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [6,3,5,1,4,2] => ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> [[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [1,5,3,6,4,2] => ? = 2 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6,3,5,4,1,2] => ? = 2 - 1
[[1,2,4],[3,6],[5]]
=> [[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [1,5,6,3,4,2] => ? = 3 - 1
[[1,4],[2,5],[3,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [5,3,2,4,1,6] => ? = 2 - 1
[[1,3],[2,5],[4,6]]
=> [[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [5,3,2,6,4,1] => ? = 2 - 1
[[1,2],[3,5],[4,6]]
=> [[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [5,3,6,2,4,1] => ? = 1 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [5,3,2,6,1,4] => ? = 2 - 1
[[1,2],[3,4],[5,6]]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [5,3,6,2,1,4] => ? = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [6,4,1,3,5,2] => ? = 2 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [6,4,1,3,2,5] => ? = 2 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [6,4,3,1,2,5] => ? = 2 - 1
[[1,2],[3,6],[4],[5]]
=> [[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [3,1,4,5,6,2] => ? = 2 - 1
[[1,4],[2,5],[3],[6]]
=> [[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [6,4,3,5,2,1] => ? = 2 - 1
[[1,3],[2,5],[4],[6]]
=> [[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [6,4,3,1,5,2] => ? = 2 - 1
[[1,2],[3,5],[4],[6]]
=> [[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [3,1,4,5,2,6] => ? = 2 - 1
[[1,3],[2,4],[5],[6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [3,1,2,6,5,4] => ? = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,1,4,2,6,5] => ? = 2 - 1
[[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,1,2,3,4] => ? = 1 - 1
[[1,3,5,6],[2,4,7]]
=> [[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [5,1,4,6,2,3,7] => ? = 1 - 1
[[1,2,5,6],[3,4,7]]
=> [[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [5,1,7,4,6,2,3] => ? = 2 - 1
[[1,4,6],[2,5,7],[3]]
=> [[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [7,4,3,6,1,5,2] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [[1,3,4],[2,5,6],[7]]
=> [7,2,5,6,1,3,4] => [7,4,3,2,6,1,5] => ? = 1 - 1
[[1,2,6],[3,5,7],[4]]
=> [[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [4,1,5,7,3,6,2] => ? = 2 - 1
[[1,3,6],[2,4,7],[5]]
=> [[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [4,1,3,6,2,7,5] => ? = 2 - 1
[[1,2,6],[3,4,7],[5]]
=> [[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [4,1,5,3,6,2,7] => ? = 2 - 1
[[1,4,5],[2,6,7],[3]]
=> [[1,2,5],[3,4,6],[7]]
=> [7,3,4,6,1,2,5] => [7,4,3,6,5,1,2] => ? = 2 - 1
[[1,3,5],[2,6,7],[4]]
=> [[1,2,4],[3,5,6],[7]]
=> [7,3,5,6,1,2,4] => [7,4,3,2,6,5,1] => ? = 2 - 1
[[1,2,5],[3,6,7],[4]]
=> [[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [4,1,5,7,3,2,6] => ? = 2 - 1
[[1,3,4],[2,6,7],[5]]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [7,4,3,2,1,6,5] => ? = 2 - 1
[[1,2,4],[3,6,7],[5]]
=> [[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [4,1,5,7,6,3,2] => ? = 2 - 1
[[1,2,3],[4,6,7],[5]]
=> [[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [4,1,7,5,6,3,2] => ? = 2 - 1
[[1,3,5],[2,4,7],[6]]
=> [[1,2,6],[3,4,7],[5]]
=> [5,3,4,7,1,2,6] => [4,1,3,2,6,7,5] => ? = 2 - 1
[[1,2,5],[3,4,7],[6]]
=> [[1,2,5],[3,4,7],[6]]
=> [6,3,4,7,1,2,5] => [4,1,5,3,2,6,7] => ? = 3 - 1
[[1,3,4],[2,5,7],[6]]
=> [[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [4,1,3,2,6,5,7] => ? = 2 - 1
[[1,2,4],[3,5,7],[6]]
=> [[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [4,1,7,3,2,6,5] => ? = 2 - 1
[[1,2,3],[4,5,7],[6]]
=> [[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [4,1,7,5,3,2,6] => ? = 2 - 1
[[1,3,5],[2,4,6],[7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [4,1,3,6,5,2,7] => ? = 1 - 1
[[1,2,5],[3,4,6],[7]]
=> [[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [4,1,7,3,6,5,2] => ? = 1 - 1
[[1,3,4],[2,5,6],[7]]
=> [[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [4,1,3,6,2,5,7] => ? = 1 - 1
[[1,2,4],[3,5,6],[7]]
=> [[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [4,1,7,3,6,2,5] => ? = 2 - 1
[[1,2,3],[4,5,6],[7]]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [4,1,7,5,3,6,2] => ? = 2 - 1
[[1,4,6],[2,5],[3,7]]
=> [[1,3,5],[2,6],[4,7]]
=> [4,7,2,6,1,3,5] => [6,3,2,5,1,4,7] => ? = 2 - 1
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001960
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Mp00069: Permutations —complement⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => [1] => [1] => ? = 0 - 1
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,4,3,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,4,1,5,2] => [2,5,1,4,3] => 1 = 2 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,5,3,1,4] => 1 = 2 - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,3,1,5,4] => [2,5,3,4,1] => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,4,2,5,3] => [2,3,5,1,4] => 0 = 1 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => [2,3,5,4,1] => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [5,3,1,6,4,2] => [3,6,2,4,1,5] => ? = 1 - 1
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [4,3,1,6,5,2] => [3,6,2,1,4,5] => ? = 1 - 1
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [5,2,1,6,4,3] => [3,6,5,2,4,1] => ? = 2 - 1
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [4,2,1,6,5,3] => [3,6,5,2,1,4] => ? = 2 - 1
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [3,6,5,4,2,1] => ? = 2 - 1
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [4,5,1,6,3,2] => [3,6,2,5,4,1] => ? = 1 - 1
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [3,5,1,6,4,2] => [3,6,2,5,1,4] => ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [3,4,1,6,5,2] => [3,6,2,1,5,4] => ? = 2 - 1
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [2,5,1,6,4,3] => [3,6,4,2,5,1] => ? = 2 - 1
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [2,4,1,6,5,3] => [3,6,4,2,1,5] => ? = 3 - 1
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [4,1,5,2,6,3] => [2,4,6,1,5,3] => ? = 2 - 1
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,1,5,2,6,4] => [2,4,6,3,1,5] => ? = 2 - 1
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,1,4,2,6,5] => [2,4,6,3,5,1] => ? = 1 - 1
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [2,1,5,3,6,4] => [2,4,3,6,1,5] => ? = 2 - 1
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => [2,4,3,6,5,1] => ? = 2 - 1
[[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => [3,4,5,1,6,2] => [2,6,1,5,4,3] => ? = 2 - 1
[[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [2,4,5,1,6,3] => [2,6,3,1,5,4] => ? = 2 - 1
[[1,3],[2,6],[4],[5]]
=> [5,4,2,6,1,3] => [2,3,5,1,6,4] => [2,6,3,4,1,5] => ? = 2 - 1
[[1,2],[3,6],[4],[5]]
=> [5,4,3,6,1,2] => [2,3,4,1,6,5] => [2,6,3,4,5,1] => ? = 2 - 1
[[1,4],[2,5],[3],[6]]
=> [6,3,2,5,1,4] => [1,4,5,2,6,3] => [2,3,6,1,5,4] => ? = 2 - 1
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [1,3,5,2,6,4] => [2,3,6,4,1,5] => ? = 2 - 1
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [1,3,4,2,6,5] => [2,3,6,4,5,1] => ? = 2 - 1
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [1,2,5,3,6,4] => [2,3,4,6,1,5] => ? = 1 - 1
[[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,2,4,3,6,5] => [2,3,4,6,5,1] => ? = 2 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [6,4,1,7,5,3,2] => [4,7,3,5,1,6,2] => ? = 1 - 1
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [5,4,1,7,6,3,2] => [4,7,3,1,5,6,2] => ? = 2 - 1
[[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => [5,6,3,1,7,4,2] => [3,7,2,5,4,1,6] => ? = 1 - 1
[[1,3,6],[2,5,7],[4]]
=> [4,2,5,7,1,3,6] => [4,6,3,1,7,5,2] => [3,7,2,5,1,4,6] => ? = 1 - 1
[[1,2,6],[3,5,7],[4]]
=> [4,3,5,7,1,2,6] => [4,5,3,1,7,6,2] => [3,7,2,1,5,4,6] => ? = 2 - 1
[[1,3,6],[2,4,7],[5]]
=> [5,2,4,7,1,3,6] => [3,6,4,1,7,5,2] => [3,7,2,5,1,6,4] => ? = 2 - 1
[[1,2,6],[3,4,7],[5]]
=> [5,3,4,7,1,2,6] => [3,5,4,1,7,6,2] => [3,7,2,1,5,6,4] => ? = 2 - 1
[[1,4,5],[2,6,7],[3]]
=> [3,2,6,7,1,4,5] => [5,6,2,1,7,4,3] => [3,7,6,2,5,4,1] => ? = 2 - 1
[[1,3,5],[2,6,7],[4]]
=> [4,2,6,7,1,3,5] => [4,6,2,1,7,5,3] => [3,7,6,2,5,1,4] => ? = 2 - 1
[[1,2,5],[3,6,7],[4]]
=> [4,3,6,7,1,2,5] => [4,5,2,1,7,6,3] => [3,7,6,2,1,5,4] => ? = 2 - 1
[[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [3,6,2,1,7,5,4] => [3,7,6,4,2,5,1] => ? = 2 - 1
[[1,2,4],[3,6,7],[5]]
=> [5,3,6,7,1,2,4] => [3,5,2,1,7,6,4] => [3,7,6,4,2,1,5] => ? = 2 - 1
[[1,2,3],[4,6,7],[5]]
=> [5,4,6,7,1,2,3] => [3,4,2,1,7,6,5] => [3,7,6,4,5,2,1] => ? = 2 - 1
[[1,3,5],[2,4,7],[6]]
=> [6,2,4,7,1,3,5] => [2,6,4,1,7,5,3] => [3,7,4,2,5,1,6] => ? = 2 - 1
[[1,2,5],[3,4,7],[6]]
=> [6,3,4,7,1,2,5] => [2,5,4,1,7,6,3] => [3,7,4,2,1,5,6] => ? = 3 - 1
[[1,3,4],[2,5,7],[6]]
=> [6,2,5,7,1,3,4] => [2,6,3,1,7,5,4] => [3,7,4,6,2,5,1] => ? = 2 - 1
[[1,2,4],[3,5,7],[6]]
=> [6,3,5,7,1,2,4] => [2,5,3,1,7,6,4] => [3,7,4,6,2,1,5] => ? = 2 - 1
[[1,2,3],[4,5,7],[6]]
=> [6,4,5,7,1,2,3] => [2,4,3,1,7,6,5] => [3,7,4,6,5,2,1] => ? = 2 - 1
[[1,3,5],[2,4,6],[7]]
=> [7,2,4,6,1,3,5] => [1,6,4,2,7,5,3] => [3,4,7,2,5,1,6] => ? = 1 - 1
[[1,2,5],[3,4,6],[7]]
=> [7,3,4,6,1,2,5] => [1,5,4,2,7,6,3] => [3,4,7,2,1,5,6] => ? = 1 - 1
[[1,3,4],[2,5,6],[7]]
=> [7,2,5,6,1,3,4] => [1,6,3,2,7,5,4] => [3,4,7,6,2,5,1] => ? = 1 - 1
[[1,2,4],[3,5,6],[7]]
=> [7,3,5,6,1,2,4] => [1,5,3,2,7,6,4] => [3,4,7,6,2,1,5] => ? = 2 - 1
[[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,4,3,2,7,6,5] => [3,4,7,6,5,2,1] => ? = 2 - 1
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [5,1,6,3,7,4,2] => [3,5,2,6,4,1,7] => ? = 2 - 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
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