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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
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(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ 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] => [1] => 1
[1,1] => [2] => 2
[2] => [1] => 1
[1,1,1] => [3] => 2
[3] => [1] => 1
[1,1,1,1] => [4] => 2
[2,1,1] => [1,2] => 3
[2,2] => [2] => 2
[4] => [1] => 1
[1,1,1,1,1] => [5] => 2
[1,2,1,1] => [1,1,2] => 3
[1,2,2] => [1,2] => 3
[2,1,1,1] => [1,3] => 3
[3,1,1] => [1,2] => 3
[5] => [1] => 1
[1,1,1,1,1,1] => [6] => 2
[1,1,2,1,1] => [2,1,2] => 4
[1,1,2,2] => [2,2] => 4
[1,2,1,1,1] => [1,1,3] => 3
[1,3,1,1] => [1,1,2] => 3
[2,1,1,1,1] => [1,4] => 3
[2,2,1,1] => [2,2] => 4
[2,2,2] => [3] => 2
[3,1,1,1] => [1,3] => 3
[3,3] => [2] => 2
[4,1,1] => [1,2] => 3
[6] => [1] => 1
[1,1,1,1,1,1,1] => [7] => 2
[1,1,1,2,1,1] => [3,1,2] => 4
[1,1,1,2,2] => [3,2] => 4
[1,1,2,1,1,1] => [2,1,3] => 4
[1,1,3,1,1] => [2,1,2] => 4
[1,2,1,1,1,1] => [1,1,4] => 3
[1,2,2,1,1] => [1,2,2] => 5
[1,2,2,2] => [1,3] => 3
[1,3,1,1,1] => [1,1,3] => 3
[1,3,3] => [1,2] => 3
[1,4,1,1] => [1,1,2] => 3
[2,1,1,1,1,1] => [1,5] => 3
[2,1,2,1,1] => [1,1,1,2] => 3
[2,1,2,2] => [1,1,2] => 3
[2,2,1,1,1] => [2,3] => 4
[2,3,1,1] => [1,1,2] => 3
[3,1,1,1,1] => [1,4] => 3
[3,2,1,1] => [1,1,2] => 3
[3,2,2] => [1,2] => 3
[4,1,1,1] => [1,3] => 3
[5,1,1] => [1,2] => 3
[7] => [1] => 1
[1,1,1,1,2,1,1] => [4,1,2] => 4
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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ 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%
Mp00039: Integer compositions —complement⟶ 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%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,1] => [2] => [1,1] => ([(0,1)],2)
=> 2
[2] => [1] => [1] => ([],1)
=> 1
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3] => [1] => [1] => ([],1)
=> 1
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,2] => [2] => [1,1] => ([(0,1)],2)
=> 2
[4] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[5] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,2,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,3] => [2] => [1,1] => ([(0,1)],2)
=> 2
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[6] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,1,1,1] => [2,1,3] => [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
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,1,1,1,1] => [1,1,4] => [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
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,2,2] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[1,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1,1,1] => [1,5] => [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
[2,1,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[4,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[7] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [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
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000340
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ 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%
Mp00173: Integer compositions —rotate front to back⟶ 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] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1] => [2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[2] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1,1] => [3] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1,1,1] => [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,2] => [2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[4] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[5] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,2,1,1] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,2,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,1,1,1] => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1,1] => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[2,2,1,1] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[2,2,2] => [3] => [3] => [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,1,1] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,3] => [2] => [2] => [1,1,0,0]
=> 1 = 2 - 1
[4,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[6] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,2,1,1] => [3,1,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,2,2] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,2,1,1,1] => [2,1,3] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,3,1,1] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,1,1,1,1] => [1,1,4] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[1,2,2,1,1] => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,2,2] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,1,1,1] => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,3,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1,1,1] => [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2 = 3 - 1
[2,1,2,1,1] => [1,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,2,2] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,2,1,1,1] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[2,3,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,1,1,1] => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[4,1,1,1] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[5,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[7] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ 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%
Mp00173: Integer compositions —rotate front to back⟶ 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] => [1] => [1] => 1 => 0 = 1 - 1
[1,1] => [2] => [2] => 10 => 1 = 2 - 1
[2] => [1] => [1] => 1 => 0 = 1 - 1
[1,1,1] => [3] => [3] => 100 => 1 = 2 - 1
[3] => [1] => [1] => 1 => 0 = 1 - 1
[1,1,1,1] => [4] => [4] => 1000 => 1 = 2 - 1
[2,1,1] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[2,2] => [2] => [2] => 10 => 1 = 2 - 1
[4] => [1] => [1] => 1 => 0 = 1 - 1
[1,1,1,1,1] => [5] => [5] => 10000 => 1 = 2 - 1
[1,2,1,1] => [1,1,2] => [1,2,1] => 1101 => 2 = 3 - 1
[1,2,2] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[2,1,1,1] => [1,3] => [3,1] => 1001 => 2 = 3 - 1
[3,1,1] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[5] => [1] => [1] => 1 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => [6] => 100000 => 1 = 2 - 1
[1,1,2,1,1] => [2,1,2] => [1,2,2] => 11010 => 3 = 4 - 1
[1,1,2,2] => [2,2] => [2,2] => 1010 => 3 = 4 - 1
[1,2,1,1,1] => [1,1,3] => [1,3,1] => 11001 => 2 = 3 - 1
[1,3,1,1] => [1,1,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,1,1,1,1] => [1,4] => [4,1] => 10001 => 2 = 3 - 1
[2,2,1,1] => [2,2] => [2,2] => 1010 => 3 = 4 - 1
[2,2,2] => [3] => [3] => 100 => 1 = 2 - 1
[3,1,1,1] => [1,3] => [3,1] => 1001 => 2 = 3 - 1
[3,3] => [2] => [2] => 10 => 1 = 2 - 1
[4,1,1] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[6] => [1] => [1] => 1 => 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => [7] => 1000000 => 1 = 2 - 1
[1,1,1,2,1,1] => [3,1,2] => [1,2,3] => 110100 => 3 = 4 - 1
[1,1,1,2,2] => [3,2] => [2,3] => 10100 => 3 = 4 - 1
[1,1,2,1,1,1] => [2,1,3] => [1,3,2] => 110010 => 3 = 4 - 1
[1,1,3,1,1] => [2,1,2] => [1,2,2] => 11010 => 3 = 4 - 1
[1,2,1,1,1,1] => [1,1,4] => [1,4,1] => 110001 => 2 = 3 - 1
[1,2,2,1,1] => [1,2,2] => [2,2,1] => 10101 => 4 = 5 - 1
[1,2,2,2] => [1,3] => [3,1] => 1001 => 2 = 3 - 1
[1,3,1,1,1] => [1,1,3] => [1,3,1] => 11001 => 2 = 3 - 1
[1,3,3] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[1,4,1,1] => [1,1,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,1,1,1,1,1] => [1,5] => [5,1] => 100001 => 2 = 3 - 1
[2,1,2,1,1] => [1,1,1,2] => [1,1,2,1] => 11101 => 2 = 3 - 1
[2,1,2,2] => [1,1,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,2,1,1,1] => [2,3] => [3,2] => 10010 => 3 = 4 - 1
[2,3,1,1] => [1,1,2] => [1,2,1] => 1101 => 2 = 3 - 1
[3,1,1,1,1] => [1,4] => [4,1] => 10001 => 2 = 3 - 1
[3,2,1,1] => [1,1,2] => [1,2,1] => 1101 => 2 = 3 - 1
[3,2,2] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[4,1,1,1] => [1,3] => [3,1] => 1001 => 2 = 3 - 1
[5,1,1] => [1,2] => [2,1] => 101 => 2 = 3 - 1
[7] => [1] => [1] => 1 => 0 = 1 - 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,2,4] => 1101000 => 3 = 4 - 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: St000453
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ 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%
Mp00039: Integer compositions —complement⟶ 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%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,1] => [2] => [1,1] => ([(0,1)],2)
=> 2
[2] => [1] => [1] => ([],1)
=> 1
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3] => [1] => [1] => ([],1)
=> 1
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,2] => [2] => [1,1] => ([(0,1)],2)
=> 2
[4] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[5] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,2,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,3] => [2] => [1,1] => ([(0,1)],2)
=> 2
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[6] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,1,1,1] => [2,1,3] => [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
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,1,1,1,1] => [1,1,4] => [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
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,2,2] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[1,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,1,1,1,1,1] => [1,5] => [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
[2,1,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[2,1,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[4,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3
[7] => [1] => [1] => ([],1)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [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
[1,1,1,2,1,1,1] => [3,1,3] => [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
[1,1,1,3,1,1,1] => [3,1,3] => [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
[1,1,1,4,1,1,1] => [3,1,3] => [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
[1,1,1,4,2,2,2] => [3,1,3] => [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
[1,1,1,6,1,1,1] => [3,1,3] => [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
[2,2,2,4,1,1,1] => [3,1,3] => [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
[2,2,2,3,1,1,1] => [3,1,3] => [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
[1,1,1,5,1,1,1] => [3,1,3] => [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
[1,1,1,7,1,1,1] => [3,1,3] => [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: St001035
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 98%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 98%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[2] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[3] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[4] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[5] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[3,3] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[6] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[3,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[5,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[7] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 4 - 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 4 = 6 - 2
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2 = 4 - 2
[8] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[9] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[10] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[12] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 2
[11] => [1] => [1,0]
=> [1,0]
=> ? = 1 - 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: St000796
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1] => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
[2] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
[3] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 2 - 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[2,2] => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
[4] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 2 - 2
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 3 - 2
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[5] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0 = 2 - 2
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 4 - 2
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 4 - 2
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1 = 3 - 2
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 3 - 2
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2 = 4 - 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 2 - 2
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 3 - 2
[3,3] => [2] => [1,1,0,0]
=> [1,2] => 0 = 2 - 2
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[6] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => 0 = 2 - 2
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 2 = 4 - 2
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 2 = 4 - 2
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 2 = 4 - 2
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2 = 4 - 2
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => 1 = 3 - 2
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 3 = 5 - 2
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 3 - 2
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1 = 3 - 2
[1,3,3] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1 = 3 - 2
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1 = 3 - 2
[2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2 = 4 - 2
[2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1 = 3 - 2
[3,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1 = 3 - 2
[3,2,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[4,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 3 - 2
[5,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 3 - 2
[7] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => 2 = 4 - 2
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 2 = 4 - 2
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => 2 = 4 - 2
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 2 = 4 - 2
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => 2 = 4 - 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => 4 = 6 - 2
[1,1,2,2,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 2 = 4 - 2
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 2 = 4 - 2
[1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 2
[2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 3 - 2
[8] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 5 - 2
[1,2,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 3 - 2
[1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 5 - 2
[1,3,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 2
[2,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 5 - 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => ? = 5 - 2
[2,1,2,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 3 - 2
[3,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 3 - 2
[9] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,2,1,1,1,2,2] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => ? = 5 - 2
[1,2,1,1,3,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 5 - 2
[1,2,1,2,2,1,1] => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => ? = 5 - 2
[1,2,1,3,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 3 - 2
[1,2,2,1,2,1,1] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 5 - 2
[1,2,2,2,1,1,1] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 5 - 2
[1,2,3,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 3 - 2
[1,3,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => ? = 5 - 2
[1,3,1,2,1,1,1] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 3 - 2
[1,3,2,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 3 - 2
[1,4,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 2
[2,1,1,1,1,2,2] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 5 - 2
[2,1,1,1,3,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 5 - 2
[2,1,1,2,2,1,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 7 - 2
[2,1,1,3,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => ? = 5 - 2
[2,1,2,1,2,1,1] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ? = 3 - 2
[2,1,2,2,1,1,1] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => ? = 5 - 2
[2,1,3,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 3 - 2
[2,3,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 2
[3,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 5 - 2
[3,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => ? = 5 - 2
[3,1,2,1,1,1,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => ? = 3 - 2
[3,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? = 3 - 2
[4,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 3 - 2
[10] => [1] => [1,0]
=> [1] => ? = 1 - 2
[1,2,2,2,2,2,2] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? = 3 - 2
[1,2,2,1,3,2,2] => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 5 - 2
[1,2,2,1,1,3,3] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 7 - 2
[1,2,1,3,2,2,2] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ? = 3 - 2
[1,2,1,2,3,2,2] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ? = 3 - 2
[1,2,1,2,1,3,3] => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ? = 3 - 2
Description
The stat' of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(\underline{13}2)$, $(\underline{31}2)$, $(\underline{32}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000388
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 86%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[2] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1] => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[4] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1] => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,2] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[5] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1,1] => [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,2,2] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,3] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[4,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[6] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1,1,1] => [7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 4
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,2,2] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,3] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1,1] => [1,5] => ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,1,2,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,2,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,2] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[7] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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
[1,1,1,1,2,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,1,2,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(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
[1,1,1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 4
[1,1,2,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,1,1,1,1,1] => [1,6] => ([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,2,2] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,3,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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
[1,1,1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(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)
=> ? = 6
[1,1,1,3,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(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
[1,1,2,1,2,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,2,2,1,1,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,3,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,2,1,1] => [1,1,2,1,2] => ([(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)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,2,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,2,1,1,1,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,3,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,1,1,2,1,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,1,2,1,1,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,2,1,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,2,1,1,1,1,1] => [2,5] => ([(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[3,1,1,1,1,1,1] => [1,6] => ([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,2,2,2] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,4,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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
[1,1,1,2,1,2,2] => [3,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,2,3,1,1] => [3,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,3,2,1,1] => [3,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,4,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(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
[1,1,2,1,1,2,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,2,1,3,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,2,2,2,1,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,2,3,1,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,3,1,2,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,3,2,1,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,4,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,1,2,2] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,1,3,1,1] => [1,1,2,1,2] => ([(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)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,2,2,1,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,3,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,2,1,2,1,1] => [1,2,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,2,2,1,1,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,3,1,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,1,1,2,1,1] => [1,1,2,1,2] => ([(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)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,3,1,2,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,2,1,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,4,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,1,1,1,2,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,1,1,3,1,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,1,2,2,1,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[2,1,1,3,1,1,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
Description
The number of orbits of vertices of a graph under automorphisms.
Matching statistic: St001951
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 86%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[2] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1] => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,2] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[4] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1] => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,2,2] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[5] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1,1] => [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[2,2,2] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,3] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2
[4,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[6] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1,1,1] => [7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 4
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[1,2,2,2] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,3,3] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1,1] => [1,5] => ([(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[2,1,2,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,1,2,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,2,1,1,1] => [2,3] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[2,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[3,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,2] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[7] => [1] => ([],1)
=> ([(0,1)],2)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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
[1,1,1,1,2,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[1,1,1,2,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(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
[1,1,1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 4
[1,1,2,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,1,1,1,1,1] => [1,6] => ([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,1,2,2] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,3,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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
[1,1,1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(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)
=> ? = 6
[1,1,1,3,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(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
[1,1,2,1,2,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,2,2,1,1,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,3,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,2,1,1] => [1,1,2,1,2] => ([(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)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,2,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,2,1,1,1,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,3,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,1,1,2,1,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,1,2,1,1,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,2,1,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,2,1,1,1,1,1] => [2,5] => ([(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[3,1,1,1,1,1,1] => [1,6] => ([(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,1,2,2,2] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,1,4,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(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
[1,1,1,2,1,2,2] => [3,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,2,3,1,1] => [3,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,3,2,1,1] => [3,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,1,4,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(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
[1,1,2,1,1,2,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,2,1,3,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,2,2,2,1,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,1,2,3,1,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,3,1,2,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,3,2,1,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,1,4,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,2,1,1,1,2,2] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,1,3,1,1] => [1,1,2,1,2] => ([(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)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,2,2,1,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,1,3,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,2,2,1,2,1,1] => [1,2,1,1,2] => ([(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)
=> ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,2,2,1,1,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,2,3,1,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,1,1,2,1,1] => [1,1,2,1,2] => ([(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)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,3,1,2,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,2,1,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,4,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[2,1,1,1,1,2,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,1,1,3,1,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[2,1,1,2,2,1,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7
[2,1,1,3,1,1,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
Description
The number of factors in the disjoint direct product decomposition of the automorphism group of a graph.
The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$.
In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 51%●distinct values known / distinct values provided: 29%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 51%●distinct values known / distinct values provided: 29%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1] => [2] => [1,1] => ([(0,1)],2)
=> -1 = 2 - 3
[2] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
[3] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,2] => [2] => [1,1] => ([(0,1)],2)
=> -1 = 2 - 3
[4] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1,1,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 2 - 3
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[5] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 2 - 3
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,1,1,1,1] => [1,4] => [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
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[2,2,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 2 - 3
[3,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,3] => [2] => [1,1] => ([(0,1)],2)
=> -1 = 2 - 3
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[6] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1 = 2 - 3
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,1,2,1,1,1] => [2,1,3] => [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
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,2,1,1,1,1] => [1,1,4] => [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
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[1,2,2,2] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[1,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,1,1,1,1,1] => [1,5] => [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
[2,1,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,1,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,1,1,1] => [1,4] => [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
[3,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[4,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[7] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1,1,2,1,1] => [4,1,2] => [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
[1,1,1,1,2,2] => [4,2] => [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
[1,1,1,2,1,1,1] => [3,1,3] => [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
[1,1,1,3,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,2,1,1,1,1] => [2,1,4] => [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
[1,1,2,2,1,1] => [2,2,2] => [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
[1,1,2,2,2] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,1,3,1,1,1] => [2,1,3] => [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
[1,1,3,3] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[1,1,4,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,2,1,1,1,1,1] => [1,1,5] => [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
[1,2,1,2,1,1] => [1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[1,2,1,2,2] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,2,2,1,1,1] => [1,2,3] => [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
[1,2,3,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,3,1,1,1,1] => [1,1,4] => [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
[1,3,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,3,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[1,4,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[1,5,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,1,1,1,1,1,1] => [1,6] => [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
[2,1,1,2,1,1] => [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 3
[2,1,1,2,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5 - 3
[2,1,2,1,1,1] => [1,1,1,3] => [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
[2,1,3,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,2,1,1,1,1] => [2,4] => [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
[2,2,2,1,1] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[2,2,2,2] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 2 - 3
[2,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0 = 3 - 3
[2,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,1,1,1,1] => [1,5] => [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
[3,3,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4 - 3
[8] => [1] => [1] => ([],1)
=> ? = 1 - 3
[1,1,1,1,1,2,2] => [5,2] => [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
[1,1,1,1,3,1,1] => [4,1,2] => [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
[1,1,1,2,2,1,1] => [3,2,2] => [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
[1,1,1,2,2,2] => [3,3] => [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
[1,1,1,3,1,1,1] => [3,1,3] => [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
[1,1,1,3,3] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,1,1,4,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 3
[1,1,2,1,2,2] => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,2,2,1,1,1] => [2,2,3] => [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
[1,1,2,3,1,1] => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,3,1,1,1,1] => [2,1,4] => [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
[1,1,3,2,1,1] => [2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 3
[1,1,3,2,2] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,1,4,1,1,1] => [2,1,3] => [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
[1,1,5,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 3
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 - 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.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001352The number of internal nodes in the modular decomposition of a graph. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001488The number of corners of a skew partition. St000831The number of indices that are either descents or recoils. St000897The number of different multiplicities of parts of an integer partition. St000670The reversal length of a permutation.
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