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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1] => 2
[1,0,1,0,1,0]
=> [1,1,1] => 2
[1,1,0,0,1,0]
=> [2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 4
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 3
[1,1,0,1,0,0,1,0]
=> [3,1] => 3
[1,1,1,0,0,0,1,0]
=> [3,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 3
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => 4
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => 5
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => 5
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => 5
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => 5
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => 3
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000777
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [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)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [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)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [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)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [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)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(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)
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [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)
=> 5
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [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)
=> 5
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000340
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000691
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 0 = 1 - 1
[1,0,1,0]
=> [1,1] => [2] => 10 => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 100 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 101 => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 1000 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 1010 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => 1001 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 10000 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => 10100 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => 10010 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 10110 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 10001 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => 100000 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,4] => 101000 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [3,3] => 100100 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => 101100 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,3] => 101100 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [4,2] => 100010 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,2] => 101010 => 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => 100110 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,2] => 100110 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [2,1,1,2] => 101110 => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [5,1] => 100001 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [2,3,1] => 101001 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [3,2,1] => 100101 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [2,1,2,1] => 101101 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [2,1,2,1] => 101101 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [4,1,1] => 100011 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [2,2,1,1] => 101011 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [3,1,1,1] => 100111 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => 101111 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => 101111 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => [3,1,1,1] => 100111 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => 101111 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => [2,1,1,1,1] => 101111 => 2 = 3 - 1
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St001035
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 99%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1 - 2
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,5,3,4,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,3,4,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => [5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [4,1,3,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => [5,1,3,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [5,1,4,3,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 3 - 2
Description
The convexity degree of the parallelogram polyomino associated with the Dyck path.
A parallelogram polyomino is $k$-convex if $k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino.
For example, any rotation of a Ferrers shape has convexity degree at most one.
The (bivariate) generating function is given in Theorem 2 of [1].
Matching statistic: St000453
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [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)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [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)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [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)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [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)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(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)
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [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)
=> 5
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [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)
=> 5
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,3,1,1] => ([(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),(5,6)],7)
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,3,1,1] => ([(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),(5,6)],7)
=> ? = 4
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000455
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 53%●distinct values known / distinct values provided: 29%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 53%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [1] => ([],1)
=> ? = 1 - 3
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> -1 = 2 - 3
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,0,1,0,1,0,1,0,1,0]
=> [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 = 2 - 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [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 = 2 - 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [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)
=> ? = 4 - 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [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)
=> ? = 4 - 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [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)
=> ? = 4 - 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(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)
=> 0 = 3 - 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [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)
=> ? = 5 - 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [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)
=> ? = 5 - 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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 = 2 - 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,2,1] => ([(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)
=> ? = 4 - 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1] => ([(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)
=> ? = 4 - 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1] => ([(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)
=> ? = 4 - 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [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)
=> ? = 6 - 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,3,1,1] => ([(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),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,3,1,1] => ([(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),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1] => ([(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)
=> ? = 4 - 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [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)
=> ? = 6 - 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [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)
=> ? = 6 - 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [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)
=> ? = 6 - 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [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)
=> ? = 6 - 3
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,1,1,1] => ([(0,4),(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)
=> ? = 4 - 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [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)
=> ? = 6 - 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1] => ([(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)
=> 0 = 3 - 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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 = 3 - 3
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 3 - 3
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 3 - 3
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: St001730
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 57%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> []
=> => ? = 1 - 2
[1,0,1,0]
=> [1]
=> 10 => 0 = 2 - 2
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0 = 2 - 2
[1,1,0,0,1,0]
=> [2]
=> 100 => 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 0 = 2 - 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 1 = 3 - 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 0 = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 2 = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 10010110 => 2 = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 2 = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => 1 = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1010010 => 1 = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1001010 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1000110 => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => 1 = 3 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 100100 => 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => ? = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 1010011010 => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 1001011010 => ? = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 1010100110 => ? = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1001100110 => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 1010010110 => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 1001010110 => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1000110110 => ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => ? = 4 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> 1001001110 => ? = 4 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> 1000101110 => ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => ? = 4 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 101010100 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 100110100 => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> 101001100 => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> 100101100 => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 100011100 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> 101010010 => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> 100110010 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> 101001010 => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> 100101010 => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1]
=> 100011010 => 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> 101000110 => 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1]
=> 100100110 => 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1]
=> 100010110 => 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1]
=> 100001110 => 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 10101000 => 1 = 3 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 10011000 => 3 = 5 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> 10100100 => 1 = 3 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 10010100 => 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,2,2]
=> 10001100 => 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 10100010 => 1 = 3 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 10010010 => 1 = 3 - 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 10001010 => 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1,1]
=> 10000110 => 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1010000 => 1 = 3 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> 1001000 => 1 = 3 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 1000100 => 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 1000010 => 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 100000 => 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> 101010101010 => ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> 100110101010 => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> 101001101010 => ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> 100101101010 => ? = 4 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> 100011101010 => ? = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> 101010011010 => ? = 4 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> 100110011010 => ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> 101001011010 => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> 100101011010 => ? = 4 - 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> 100011011010 => ? = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> 101000111010 => ? = 4 - 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> 100100111010 => ? = 4 - 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> 100010111010 => ? = 4 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> 100001111010 => ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> 101010100110 => ? = 4 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> 100110100110 => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> 101001100110 => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> 100101100110 => ? = 6 - 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> 100011100110 => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> 101010010110 => ? = 4 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> 100110010110 => ? = 6 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> 101001010110 => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> 100101010110 => ? = 4 - 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1,1]
=> 100011010110 => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1,1]
=> 101000110110 => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1,1]
=> 100100110110 => ? = 4 - 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1,1]
=> 100010110110 => ? = 4 - 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> 100001110110 => ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> 101010001110 => ? = 4 - 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> 100110001110 => ? = 6 - 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1,1]
=> 101001001110 => ? = 4 - 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1,1]
=> 100101001110 => ? = 4 - 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> 100011001110 => ? = 4 - 2
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,1,1]
=> 101000101110 => ? = 4 - 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1,1]
=> 100100101110 => ? = 4 - 2
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 86%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 4
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 5
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 5
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 5
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 3
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 3
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 3
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [2,3,4,7,6,1,8,5] => ([(0,7),(1,7),(2,7),(3,4),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,7,1,4,5,8,6] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,3,7,1,6,4,8,5] => ([(0,4),(1,4),(2,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [2,3,6,5,1,7,8,4] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [2,3,7,5,1,4,8,6] => ([(0,6),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [2,3,7,6,1,4,8,5] => ([(0,5),(1,5),(2,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [2,3,7,5,6,1,8,4] => ([(0,7),(1,7),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,7,3,5,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,1,7,6,3,8,5] => ([(0,7),(1,3),(2,3),(2,6),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 6
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,7,4,8,6] => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,6,1,3,4,7,8,5] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,7,1,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,7,1,3,6,4,8,5] => ([(0,7),(1,5),(2,4),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,6,1,5,3,7,8,4] => ([(0,6),(1,6),(2,3),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,7,1,5,3,4,8,6] => ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,7,1,6,3,4,8,5] => ([(0,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,7,1,5,6,3,8,4] => ([(0,6),(1,2),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,5,4,1,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,4),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,5,4,1,7,3,8,6] => ([(0,7),(1,3),(2,3),(2,6),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 6
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [2,6,4,1,3,7,8,5] => ([(0,4),(1,4),(2,5),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,7,4,1,3,5,8,6] => ([(0,7),(1,5),(2,4),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,7,4,1,6,3,8,5] => ([(0,6),(1,5),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,6,5,1,3,7,8,4] => ([(0,5),(1,5),(2,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [2,7,5,1,3,4,8,6] => ([(0,4),(1,5),(2,6),(2,7),(3,6),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [2,6,7,1,3,4,8,5] => ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [2,7,5,1,6,3,8,4] => ([(0,6),(1,4),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,6,4,5,1,7,8,3] => ([(0,7),(1,7),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [2,7,4,5,1,3,8,6] => ([(0,6),(1,2),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [2,7,4,6,1,3,8,5] => ([(0,6),(1,4),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,7,6,5,1,3,8,4] => ([(0,4),(1,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,7,4,5,6,1,8,3] => ([(0,7),(1,6),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,4,5,7,2,8,6] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> ? = 5
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> ? = 5
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,4,7,6,2,8,5] => ([(0,7),(1,4),(2,3),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> ? = 5
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 7
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 5
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 33%●distinct values known / distinct values provided: 29%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 33%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [1] => [] => ([],0)
=> ? = 1 - 2
[1,0,1,0]
=> [1,2] => [1] => ([],1)
=> 0 = 2 - 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,2] => ([],2)
=> ? = 2 - 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> ? = 2 - 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => ([(1,2)],3)
=> ? = 4 - 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> ? = 3 - 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => ([],4)
=> ? = 2 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
=> ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
=> ? = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 4 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
=> ? = 3 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 3 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5] => ([],5)
=> ? = 2 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,5,3,4,2,6] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 3 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 5 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [3,2,4,1,5,6] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [3,2,4,5,1,6] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,1,6] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,2,3,1,5,6] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,2,3,5,1,6] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,3,4,1,6] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,2,4,3,1,6] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,3,2,4,1,6] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [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 = 3 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([],6)
=> ? = 2 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? = 4 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? = 4 - 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? = 4 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? = 4 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? = 4 - 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,1,7] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,5,1,7] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,4,6,1,7] => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,3,6,4,5,1,7] => [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,5,4,1,7] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [2,4,3,5,6,1,7] => [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,3,6,5,1,7] => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,3,4,6,1,7] => [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,6,3,4,5,1,7] => [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,5,4,3,6,1,7] => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,6,4,3,5,1,7] => [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,4,5,3,1,7] => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,6,5,4,3,1,7] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,2,4,5,6,1,7] => [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [3,2,4,6,5,1,7] => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,2,5,4,6,1,7] => [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,2,6,4,5,1,7] => [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,2,3,5,6,1,7] => [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [4,2,3,6,5,1,7] => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [5,2,3,4,6,1,7] => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,2,3,4,5,1,7] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,5,4,1,7] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [5,2,4,3,6,1,7] => [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,2,4,3,5,1,7] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,4,5,3,1,7] => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 3 - 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
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