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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000454
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Mp00277: Permutations —catalanization⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[2,4,5,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,1,4] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,5,4,1,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,1,2,5,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,5,3,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,5,3,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,1,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,3,1] => [5,4,3,2,1] => [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)
=> 4
[4,5,3,1,2] => [5,4,3,2,1] => [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)
=> 4
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,3,4,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,4,2,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,2,3,4,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001270
Mp00277: Permutations —catalanization⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[2,4,5,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,1,4] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,5,4,1,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,1,2,5,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,5,3,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,5,3,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,1,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,3,1] => [5,4,3,2,1] => [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)
=> 4
[4,5,3,1,2] => [5,4,3,2,1] => [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)
=> 4
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,3,4,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,4,2,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,2,3,4,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,7,5,4,3,1,6] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[3,5,7,4,1,2,6] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[3,6,7,4,1,2,5] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[3,7,6,4,2,1,5] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[4,6,5,2,1,3,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,2,7,3,4,1,6] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,4,6,1,3,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,3,4,2,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,4,2,3,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,4,3,1,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,6,7,3,1,4,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,6,7,4,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,7,6,3,2,4,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,7,6,4,2,1,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,4,5,2,3,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,4,5,3,1,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,4,7,5,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,5,3,4,1,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,5,7,2,4,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,5,7,4,1,3,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,5,4,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,4,5,6,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,4,6,5,2,1,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,4,6,1,3,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,6,4,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,6,4,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,4,5,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,5,3,4,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
Description
The bandwidth of a graph.
The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be
ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$.
We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Matching statistic: St001962
Mp00277: Permutations —catalanization⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001962: Graphs ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001962: Graphs ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,2,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[2,4,5,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,1,4] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,5,4,1,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,1,2,5,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,5,3,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,5,3,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,1,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,3,1] => [5,4,3,2,1] => [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)
=> 4
[4,5,3,1,2] => [5,4,3,2,1] => [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)
=> 4
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,3,4,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,4,2,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,2,3,4,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,7,5,4,3,1,6] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[3,5,7,4,1,2,6] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[3,6,7,4,1,2,5] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[3,7,6,4,2,1,5] => [7,6,5,4,3,1,2] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[4,6,5,2,1,3,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,2,7,3,4,1,6] => [7,6,5,4,2,1,3] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,4,6,1,3,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,3,4,2,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,4,2,3,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,4,3,1,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,6,7,3,1,4,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,6,7,4,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,7,6,3,2,4,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[5,7,6,4,2,1,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,4,5,2,3,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,4,5,3,1,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,4,7,5,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,5,3,4,1,2,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [1,1,1,1,1,2] => ([(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)
=> ? = 5
[6,5,7,2,4,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,5,7,4,1,3,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[6,7,5,4,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,4,5,6,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,4,6,5,2,1,3] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,4,6,1,3,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,6,4,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,5,6,4,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,4,5,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,5,3,4,1,2] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [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)
=> ? = 6
Description
The proper pathwidth of a graph.
The proper pathwidth $\operatorname{ppw}(G)$ was introduced in [1] as the minimum width of a proper-path-decomposition. Barioli et al. [2] showed that if $G$ has at least one edge, then $\operatorname{ppw}(G)$ is the minimum $k$ for which $G$ is a minor of the Cartesian product $K_k \square P$ of a complete graph on $k$ vertices with a path; and further that $\operatorname{ppw}(G)$ is the minor monotone floor $\lfloor \operatorname{Z} \rfloor(G) := \min\{\operatorname{Z}(H) \mid G \preceq H\}$ of the [[St000482|zero forcing number]] $\operatorname{Z}(G)$. It can be shown [3, Corollary 9.130] that only the spanning supergraphs need to be considered for $H$ in this definition, i.e. $\lfloor \operatorname{Z} \rfloor(G) = \min\{\operatorname{Z}(H) \mid G \le H,\; V(H) = V(G)\}$.
The minimum degree $\delta$, treewidth $\operatorname{tw}$, and pathwidth $\operatorname{pw}$ satisfy
$$\delta \le \operatorname{tw} \le \operatorname{pw} \le \operatorname{ppw} = \lfloor \operatorname{Z} \rfloor \le \operatorname{pw} + 1.$$
Note that [4] uses a different notion of proper pathwidth, which is equal to bandwidth.
Matching statistic: St001644
Mp00277: Permutations —catalanization⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,4,2,5,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,5,3,4,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,5,4,2,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[2,4,5,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[2,5,3,1,4] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[3,1,4,5,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,1,5,4,2] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[3,5,4,1,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,1,2,5,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[4,1,5,3,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,2,5,3,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,5,2,1,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,5,2,3,1] => [5,4,3,2,1] => [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)
=> 4
[4,5,3,1,2] => [5,4,3,2,1] => [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)
=> 4
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[5,1,2,4,3] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,1,3,4,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,1,4,2,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,2,3,4,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,3,1,4,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,3,4,1,2] => [5,4,3,2,1] => [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)
=> 4
[5,4,1,2,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,4,3,2,1] => [5,4,3,2,1] => [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)
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,4,6,2,5] => [1,3,6,5,2,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,5,4,2,6] => [1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,5,6,2,4] => [1,4,6,5,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,3,6,5,2,4] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,2,5,3,6] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,2,6,3,5] => [1,3,6,5,2,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,5,3,2,6] => [1,4,5,3,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,4,6,3,2,5] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,2,3,4,6] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,2,4,3,6] => [1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,5,3,4,2,6] => [1,4,5,3,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,5,4,2,3,6] => [1,4,5,3,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,5] => ([(4,5)],6)
=> 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,4,6,1,5] => [2,3,6,5,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,3,5,4,1,6] => [2,3,5,4,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,3,5,6,1,4] => [2,4,6,5,1,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,3,6,5,1,4] => [2,4,5,6,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,4,1,3,5,6] => [4,3,1,2,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[2,4,5,3,1,6] => [2,4,5,3,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,4,5,6,1,3] => [3,4,6,5,1,2] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,4,6,1,3,5] => [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,4,6,3,1,5] => [2,4,5,6,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,4,6,5,1,3] => [3,4,5,6,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,5,3,1,4,6] => [5,4,3,1,2,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,5,3,4,1,6] => [2,4,5,3,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[2,6,5,4,1,3] => [3,5,6,4,1,2] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,4,6,2,5] => [2,3,6,5,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,5,4,2,6] => [2,3,5,4,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,1,5,6,2,4] => [2,4,6,5,1,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,4,5,2,1,6] => [3,4,5,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[3,5,4,1,2,6] => [3,4,5,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,1,2,6,3,5] => [2,3,6,5,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,1,5,3,2,6] => [2,4,5,3,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,2,5,3,1,6] => [3,4,5,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[4,5,2,1,3,6] => [3,4,5,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,1,2,4,3,6] => [2,3,5,4,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,1,3,4,2,6] => [2,4,5,3,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[5,1,4,2,3,6] => [2,4,5,3,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St001645
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,4,2,5,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,2,4,3] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,3,4,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,5,4,2,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,4,5,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,5,3,1,4] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[2,5,3,4,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,1,4,5,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,5,4,2] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[3,5,4,1,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,1,2,5,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,3,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,2,5,3,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,5,2,1,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [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)
=> 5 = 4 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [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)
=> 5 = 4 + 1
[5,1,2,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,2,4,3] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,1,3,4,2] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,1,4,2,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,2,3,4,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,3,1,4,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,3,4,1,2] => [5,4,3,2,1] => [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)
=> 5 = 4 + 1
[5,4,1,2,3] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [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)
=> 5 = 4 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[4,5,6,1,2,3] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[4,6,5,2,1,3] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[5,4,6,1,3,2] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[5,6,3,4,2,1] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[5,6,4,2,3,1] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[5,6,4,3,1,2] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[6,4,5,2,3,1] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[6,4,5,3,1,2] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[6,5,3,4,1,2] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [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)
=> 6 = 5 + 1
[5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[5,6,7,3,1,4,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[5,6,7,4,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[5,7,6,3,2,4,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[5,7,6,4,2,1,3] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,4,7,5,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,5,7,2,4,3,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,5,7,4,1,3,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[6,7,5,4,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,4,5,6,1,2,3] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,4,6,5,2,1,3] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,5,4,6,1,3,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,5,6,4,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,5,6,4,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,6,4,5,3,1,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,6,5,3,4,1,2] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [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)
=> 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0
[1,2] => [1,2] => [2,1] => ([],2)
=> ? = 0
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> ? = 1
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> ? = 0
[2,1,3] => [2,1,3] => [3,1,2] => ([(1,2)],3)
=> ? = 1
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? = 0
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> ? = 1
[2,4,1,3] => [4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 2
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ? = 3
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 2
[1,3,5,4,2] => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 3
[1,4,2,5,3] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 2
[1,4,5,3,2] => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 3
[1,5,2,3,4] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 2
[1,5,2,4,3] => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 3
[1,5,3,4,2] => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 3
[1,5,4,2,3] => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 3
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> ? = 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2
[2,3,5,4,1] => [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 3
[2,4,1,3,5] => [4,3,1,2,5] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ? = 2
[2,4,5,3,1] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3
[2,5,3,1,4] => [5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[2,5,3,4,1] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3
[3,1,4,5,2] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2
[3,1,5,4,2] => [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 3
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? = 2
[3,4,1,2,5] => [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 3
[3,4,5,2,1] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[3,5,4,1,2] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[4,1,2,5,3] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2
[4,1,5,3,2] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3
[4,2,5,3,1] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[4,3,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 3
[4,5,2,1,3] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[5,1,2,3,4] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2
[5,1,2,4,3] => [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 3
[5,1,3,4,2] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3
[5,1,4,2,3] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3
[5,2,3,4,1] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[5,3,1,4,2] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[5,3,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[5,4,1,2,3] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> ? = 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
=> ? = 2
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
=> ? = 2
[4,5,6,1,2,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[4,6,5,2,1,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,4,6,1,3,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,6,4,2,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,6,4,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[6,4,5,2,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[6,4,5,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[6,5,3,4,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[5,6,7,3,1,4,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[5,6,7,4,1,2,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[5,7,6,3,2,4,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[5,7,6,4,2,1,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,4,7,5,1,2,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,5,7,2,4,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,5,7,4,1,3,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[6,7,5,4,3,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,4,5,6,1,2,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,4,6,5,2,1,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,5,4,6,1,3,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,5,6,4,2,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,5,6,4,3,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,6,4,5,3,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,6,5,3,4,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => [2,1] => ([],2)
=> ? = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> ? = 1 + 1
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> ? = 0 + 1
[2,1,3] => [2,1,3] => [3,1,2] => ([(1,2)],3)
=> ? = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? = 0 + 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> ? = 1 + 1
[2,4,1,3] => [4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 2 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ? = 3 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 2 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 3 + 1
[1,4,2,5,3] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 2 + 1
[1,4,5,3,2] => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 3 + 1
[1,5,2,3,4] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 2 + 1
[1,5,2,4,3] => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 3 + 1
[1,5,3,4,2] => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 3 + 1
[1,5,4,2,3] => [1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> ? = 1 + 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2 + 1
[2,3,5,4,1] => [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 3 + 1
[2,4,1,3,5] => [4,3,1,2,5] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ? = 2 + 1
[2,4,5,3,1] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3 + 1
[2,5,3,1,4] => [5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 + 1
[2,5,3,4,1] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3 + 1
[3,1,4,5,2] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2 + 1
[3,1,5,4,2] => [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 3 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 3 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[3,5,4,1,2] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[4,1,2,5,3] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2 + 1
[4,1,5,3,2] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3 + 1
[4,2,5,3,1] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? = 3 + 1
[4,5,2,1,3] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[5,1,2,3,4] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? = 2 + 1
[5,1,2,4,3] => [2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? = 3 + 1
[5,1,3,4,2] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3 + 1
[5,1,4,2,3] => [2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? = 3 + 1
[5,2,3,4,1] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[5,3,1,4,2] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[5,3,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[5,4,1,2,3] => [3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> ? = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(3,4),(3,5)],6)
=> ? = 2 + 1
[4,5,6,1,2,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,6,5,2,1,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,4,6,1,3,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,6,4,2,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,6,4,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[6,4,5,2,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[6,4,5,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[6,5,3,4,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[5,6,7,2,3,4,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[5,6,7,3,1,4,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[5,6,7,4,1,2,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[5,7,6,3,2,4,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[5,7,6,4,2,1,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,4,7,5,1,2,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,5,7,2,4,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,5,7,4,1,3,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,7,4,5,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,7,5,3,4,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,7,5,4,2,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,7,5,4,3,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,4,5,6,1,2,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,4,6,5,2,1,3] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,5,4,6,1,3,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,5,6,3,4,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,5,6,4,2,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,5,6,4,3,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,6,4,5,2,3,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,6,4,5,3,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,6,5,3,4,1,2] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001861
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 71%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001861: Signed permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => [1,3,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 3
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 2
[1,2,4,5,3] => [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 2
[1,2,5,3,4] => [1,5,2,3,4] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 2
[1,2,5,4,3] => [5,4,1,2,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 3
[1,3,4,5,2] => [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[1,3,5,4,2] => [5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 3
[1,4,2,5,3] => [4,1,5,2,3] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 2
[1,4,5,3,2] => [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 3
[1,5,2,3,4] => [1,2,5,3,4] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2
[1,5,2,4,3] => [5,1,4,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[1,5,3,4,2] => [5,1,3,2,4] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 3
[2,1,3,4,5] => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[2,3,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 2
[2,3,5,4,1] => [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3
[2,4,1,3,5] => [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 2
[2,4,5,3,1] => [4,2,1,3,5] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 3
[2,5,3,1,4] => [2,5,3,1,4] => [4,1,3,5,2] => [4,1,3,5,2] => ? = 3
[2,5,3,4,1] => [5,2,3,1,4] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3
[3,1,4,5,2] => [3,1,4,5,2] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[3,1,5,4,2] => [5,3,1,4,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 3
[3,2,1,4,5] => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[3,4,1,2,5] => [3,4,5,1,2] => [1,4,2,5,3] => [1,4,2,5,3] => 3
[3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 3
[3,5,4,1,2] => [3,5,4,1,2] => [1,4,5,3,2] => [1,4,5,3,2] => 3
[4,1,2,5,3] => [4,1,2,5,3] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 2
[4,1,5,3,2] => [4,3,1,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 3
[4,2,5,3,1] => [4,2,1,5,3] => [2,5,3,1,4] => [2,5,3,1,4] => ? = 3
[4,3,2,1,5] => [4,5,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[4,5,2,1,3] => [4,5,2,1,3] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 3
[4,5,2,3,1] => [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,5,3] => ? = 4
[4,5,3,1,2] => [4,5,3,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 4
[5,1,2,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[5,1,2,4,3] => [5,1,2,4,3] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3
[5,1,3,4,2] => [5,1,3,4,2] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 3
[5,1,4,2,3] => [1,5,2,4,3] => [5,3,4,1,2] => [5,3,4,1,2] => ? = 3
[5,2,3,4,1] => [5,2,3,4,1] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3
[5,3,1,4,2] => [3,1,5,4,2] => [1,5,4,2,3] => [1,5,4,2,3] => 3
[5,3,4,1,2] => [3,5,1,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 4
[5,4,1,2,3] => [1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 3
[5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4
[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
[1,2,3,5,4,6] => [1,5,2,3,4,6] => [3,4,5,1,2,6] => [3,4,5,1,2,6] => ? = 2
[1,2,4,5,3,6] => [1,4,2,3,5,6] => [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 2
[1,2,4,6,3,5] => [4,6,1,2,3,5] => [3,4,1,5,6,2] => [3,4,1,5,6,2] => ? = 3
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 2
[1,2,5,4,3,6] => [1,5,4,2,3,6] => [4,5,3,1,2,6] => [4,5,3,1,2,6] => ? = 3
[1,3,4,5,2,6] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 2
[1,3,4,6,2,5] => [3,6,1,2,4,5] => [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? = 3
[1,3,5,4,2,6] => [1,5,3,2,4,6] => [4,3,5,1,2,6] => [4,3,5,1,2,6] => ? = 3
[1,3,5,6,2,4] => [3,5,1,2,4,6] => [3,1,4,5,2,6] => [3,1,4,5,2,6] => ? = 3
[1,3,6,5,2,4] => [3,6,5,1,2,4] => [1,4,5,6,3,2] => [1,4,5,6,3,2] => ? = 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [5,3,1,2,4,6] => [5,3,1,2,4,6] => ? = 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [5,2,6,1,3,4] => [5,2,6,1,3,4] => ? = 3
[1,4,5,3,2,6] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 3
[1,4,6,3,2,5] => [4,6,3,1,2,5] => [4,1,5,3,6,2] => [4,1,5,3,6,2] => ? = 2
[1,5,2,3,4,6] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 2
Description
The number of Bruhat lower covers of a permutation.
This is, for a signed permutation $\pi$, the number of signed permutations $\tau$ having a reduced word which is obtained by deleting a letter from a reduced word from $\pi$.
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