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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000340
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Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ 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%
Mp00071: Permutations —descent composition⟶ 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
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,4,3,2] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[2,3,1,4] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,4,3,1] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[3,2,1,4] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,4,2,1] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[4,2,1,3] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3
[4,3,1,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
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
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ 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%
Mp00071: Permutations —descent composition⟶ 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
[1,2] => [2,1] => [1,1] => 11 => 0
[2,1] => [1,2] => [2] => 10 => 1
[1,2,3] => [3,2,1] => [1,1,1] => 111 => 0
[1,3,2] => [3,1,2] => [1,2] => 110 => 1
[2,1,3] => [2,3,1] => [2,1] => 101 => 2
[2,3,1] => [2,1,3] => [1,2] => 110 => 1
[3,1,2] => [1,3,2] => [2,1] => 101 => 2
[3,2,1] => [1,2,3] => [3] => 100 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1111 => 0
[1,2,4,3] => [4,3,1,2] => [1,1,2] => 1110 => 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 2
[1,3,4,2] => [4,2,1,3] => [1,1,2] => 1110 => 1
[1,4,2,3] => [4,1,3,2] => [1,2,1] => 1101 => 2
[1,4,3,2] => [4,1,2,3] => [1,3] => 1100 => 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => 1011 => 2
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 3
[2,3,1,4] => [3,2,4,1] => [1,2,1] => 1101 => 2
[2,3,4,1] => [3,2,1,4] => [1,1,2] => 1110 => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 2
[2,4,3,1] => [3,1,2,4] => [1,3] => 1100 => 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => 1011 => 2
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 3
[3,2,1,4] => [2,3,4,1] => [3,1] => 1001 => 2
[3,2,4,1] => [2,3,1,4] => [2,2] => 1010 => 3
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1101 => 2
[3,4,2,1] => [2,1,3,4] => [1,3] => 1100 => 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 2
[4,1,3,2] => [1,4,2,3] => [2,2] => 1010 => 3
[4,2,1,3] => [1,3,4,2] => [3,1] => 1001 => 2
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 3
[4,3,1,2] => [1,2,4,3] => [3,1] => 1001 => 2
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 11110 => 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 2
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 11110 => 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 11101 => 2
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 11100 => 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 2
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 11010 => 3
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 2
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 11110 => 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 11101 => 2
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => 11100 => 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 2
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 11010 => 3
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 2
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => 11010 => 3
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 11101 => 2
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: St000452
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000452: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000452: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 3 = 2 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 3 = 2 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 3 = 2 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 3 = 2 + 1
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(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)
=> ?
=> ? = 1 + 1
[8,6,7,5,4,3,2,1] => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(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)
=> ?
=> ? = 3 + 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(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)
=> ?
=> ? = 3 + 1
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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)
=> ?
=> ? = 1 + 1
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[8,5,6,7,4,3,2,1] => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[6,7,5,8,4,3,2,1] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[7,5,6,8,4,3,2,1] => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[6,5,7,8,4,3,2,1] => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(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
[7,6,8,4,5,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(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
[6,7,8,4,5,3,2,1] => [3,2,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,5,4,6,3,2,1] => [2,1,2,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,4,5,6,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(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)
=> ?
=> ? = 3 + 1
[8,4,5,6,7,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[6,7,5,4,8,3,2,1] => [2,1,2,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[7,5,6,4,8,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(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
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => ([(0,5),(0,6),(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
[5,6,7,4,8,3,2,1] => [3,2,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[6,7,4,5,8,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(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)
=> ?
=> ? = 3 + 1
[7,4,5,6,8,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[5,6,4,7,8,3,2,1] => [2,3,1,1,1] => ([(0,5),(0,6),(0,7),(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)
=> ?
=> ? = 3 + 1
[6,4,5,7,8,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[5,4,6,7,8,3,2,1] => [1,4,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 3 + 1
[4,5,6,7,8,3,2,1] => [5,1,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => ([(0,6),(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
[7,6,8,5,3,4,2,1] => [1,2,1,2,1,1] => ([(0,6),(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
[6,7,8,5,3,4,2,1] => [3,1,2,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,5,6,3,4,2,1] => [2,2,2,1,1] => ([(0,6),(0,7),(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)
=> ?
=> ? = 5 + 1
[8,5,6,7,3,4,2,1] => [1,3,2,1,1] => ([(0,6),(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 + 1
[6,7,5,8,3,4,2,1] => [2,2,2,1,1] => ([(0,6),(0,7),(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)
=> ?
=> ? = 5 + 1
[7,5,6,8,3,4,2,1] => ? => ?
=> ?
=> ? = 5 + 1
[6,5,7,8,3,4,2,1] => [1,3,2,1,1] => ([(0,6),(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 + 1
[5,6,7,8,3,4,2,1] => [4,2,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,6,4,3,5,2,1] => [2,1,1,2,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[8,6,7,4,3,5,2,1] => [1,2,1,2,1,1] => ([(0,6),(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
[7,6,8,4,3,5,2,1] => [1,2,1,2,1,1] => ([(0,6),(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
[6,7,8,4,3,5,2,1] => [3,1,2,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,6,3,4,5,2,1] => [2,1,3,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[8,6,7,3,4,5,2,1] => [1,2,3,1,1] => ([(0,6),(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)
=> ?
=> ? = 5 + 1
[7,6,8,3,4,5,2,1] => [1,2,3,1,1] => ([(0,6),(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)
=> ?
=> ? = 5 + 1
[6,7,8,3,4,5,2,1] => [3,3,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [2,1,1,2,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,4,5,3,6,2,1] => ? => ?
=> ?
=> ? = 5 + 1
[7,8,5,3,4,6,2,1] => [2,1,3,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[7,8,4,3,5,6,2,1] => [2,1,3,1,1] => ([(0,6),(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)
=> ?
=> ? = 3 + 1
[8,5,6,4,3,7,2,1] => [1,2,1,2,1,1] => ([(0,6),(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
[8,4,5,6,3,7,2,1] => ? => ?
=> ?
=> ? = 5 + 1
Description
The number of distinct eigenvalues of a graph.
Matching statistic: St000453
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 88%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [2] => [2] => ([],2)
=> 1 = 0 + 1
[2,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [3] => [3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,1] => [2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => [4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,4,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,4,3] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,3,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,4,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,1,4] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,4,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,1,3] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,3,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => [5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,4,5,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,2,5,4] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,4,2,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,4,5,2] => [4,1] => [1,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,2,5,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,1,4,7,6,5] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[3,2,1,5,7,6,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[3,2,1,6,7,5,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,2,1,3,7,6,5] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,2,1,5,7,6,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,2,1,6,7,5,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,3,1,2,7,6,5] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,3,1,5,7,6,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,3,1,6,7,5,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,3,2,5,7,6,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[4,3,2,6,7,5,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,2,1,3,7,6,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,2,1,4,7,6,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,2,1,6,7,4,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,3,1,2,7,6,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,3,1,4,7,6,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,3,1,6,7,4,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,3,2,4,7,6,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,3,2,6,7,4,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,4,1,2,7,6,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,4,1,3,7,6,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,4,1,6,7,3,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,4,2,3,7,6,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,4,2,6,7,3,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[5,4,3,6,7,2,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,2,1,3,7,5,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,2,1,4,7,5,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,2,1,5,7,4,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,3,1,2,7,5,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,3,1,4,7,5,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,3,1,5,7,4,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,3,2,4,7,5,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,3,2,5,7,4,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,4,1,2,7,5,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,4,1,3,7,5,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,4,1,5,7,3,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,4,2,3,7,5,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,4,2,5,7,3,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,4,3,5,7,2,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,5,1,2,7,4,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,5,1,3,7,4,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,5,1,4,7,3,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,5,2,3,7,4,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,5,2,4,7,3,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[6,5,3,4,7,2,1] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[7,2,1,3,6,5,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[7,2,1,4,6,5,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[7,2,1,5,6,4,3] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[7,3,1,2,6,5,4] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
[7,3,1,4,6,5,2] => [1,1,3,1,1] => [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)
=> ? = 3 + 1
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000777
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,2] => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,3] => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[3,5,2,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000638
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000638: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 75%
Values
[1] => 1 = 0 + 1
[1,2] => 1 = 0 + 1
[2,1] => 2 = 1 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 3 = 2 + 1
[2,3,1] => 2 = 1 + 1
[3,1,2] => 3 = 2 + 1
[3,2,1] => 2 = 1 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 3 = 2 + 1
[1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => 2 = 1 + 1
[2,1,3,4] => 3 = 2 + 1
[2,1,4,3] => 4 = 3 + 1
[2,3,1,4] => 3 = 2 + 1
[2,3,4,1] => 2 = 1 + 1
[2,4,1,3] => 3 = 2 + 1
[2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => 4 = 3 + 1
[3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => 4 = 3 + 1
[3,4,1,2] => 3 = 2 + 1
[3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => 3 = 2 + 1
[4,1,3,2] => 4 = 3 + 1
[4,2,1,3] => 3 = 2 + 1
[4,2,3,1] => 4 = 3 + 1
[4,3,1,2] => 3 = 2 + 1
[4,3,2,1] => 2 = 1 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => 3 = 2 + 1
[1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => 3 = 2 + 1
[1,2,5,4,3] => 2 = 1 + 1
[1,3,2,4,5] => 3 = 2 + 1
[1,3,2,5,4] => 4 = 3 + 1
[1,3,4,2,5] => 3 = 2 + 1
[1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => 3 = 2 + 1
[1,3,5,4,2] => 2 = 1 + 1
[1,4,2,3,5] => 3 = 2 + 1
[1,4,2,5,3] => 4 = 3 + 1
[1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => 4 = 3 + 1
[1,4,5,2,3] => 3 = 2 + 1
[1,2,3,4,5,6,7] => ? = 0 + 1
[1,2,3,4,5,7,6] => ? = 1 + 1
[1,2,3,4,6,5,7] => ? = 2 + 1
[1,2,3,4,6,7,5] => ? = 1 + 1
[1,2,3,4,7,5,6] => ? = 2 + 1
[1,2,3,4,7,6,5] => ? = 1 + 1
[1,2,3,5,4,6,7] => ? = 2 + 1
[1,2,3,5,4,7,6] => ? = 3 + 1
[1,2,3,5,6,4,7] => ? = 2 + 1
[1,2,3,5,6,7,4] => ? = 1 + 1
[1,2,3,5,7,4,6] => ? = 2 + 1
[1,2,3,5,7,6,4] => ? = 1 + 1
[1,2,3,6,4,5,7] => ? = 2 + 1
[1,2,3,6,4,7,5] => ? = 3 + 1
[1,2,3,6,5,4,7] => ? = 2 + 1
[1,2,3,6,5,7,4] => ? = 3 + 1
[1,2,3,6,7,4,5] => ? = 2 + 1
[1,2,3,6,7,5,4] => ? = 1 + 1
[1,2,3,7,4,5,6] => ? = 2 + 1
[1,2,3,7,4,6,5] => ? = 3 + 1
[1,2,3,7,5,4,6] => ? = 2 + 1
[1,2,3,7,5,6,4] => ? = 3 + 1
[1,2,3,7,6,4,5] => ? = 2 + 1
[1,2,3,7,6,5,4] => ? = 1 + 1
[1,2,4,3,5,6,7] => ? = 2 + 1
[1,2,4,3,5,7,6] => ? = 3 + 1
[1,2,4,3,6,5,7] => ? = 4 + 1
[1,2,4,3,6,7,5] => ? = 3 + 1
[1,2,4,3,7,5,6] => ? = 4 + 1
[1,2,4,3,7,6,5] => ? = 3 + 1
[1,2,4,5,3,6,7] => ? = 2 + 1
[1,2,4,5,3,7,6] => ? = 3 + 1
[1,2,4,5,6,3,7] => ? = 2 + 1
[1,2,4,5,6,7,3] => ? = 1 + 1
[1,2,4,5,7,3,6] => ? = 2 + 1
[1,2,4,5,7,6,3] => ? = 1 + 1
[1,2,4,6,3,5,7] => ? = 2 + 1
[1,2,4,6,3,7,5] => ? = 3 + 1
[1,2,4,6,5,3,7] => ? = 2 + 1
[1,2,4,6,5,7,3] => ? = 3 + 1
[1,2,4,6,7,3,5] => ? = 2 + 1
[1,2,4,6,7,5,3] => ? = 1 + 1
[1,2,4,7,3,5,6] => ? = 2 + 1
[1,2,4,7,3,6,5] => ? = 3 + 1
[1,2,4,7,5,3,6] => ? = 2 + 1
[1,2,4,7,5,6,3] => ? = 3 + 1
[1,2,4,7,6,3,5] => ? = 2 + 1
[1,2,4,7,6,5,3] => ? = 1 + 1
[1,2,5,3,4,6,7] => ? = 2 + 1
[1,2,5,3,4,7,6] => ? = 3 + 1
Description
The number of up-down runs of a permutation.
An '''up-down run''' of a permutation $\pi=\pi_{1}\pi_{2}\cdots\pi_{n}$ is either a maximal monotone consecutive subsequence or $\pi_{1}$ if 1 is a descent of $\pi$.
For example, the up-down runs of $\pi=85712643$ are $8$, $85$, $57$, $71$, $126$, and
$643$.
Matching statistic: St000455
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
=> ([],1)
=> ? = 0 - 2
[1,2] => [2] => ([],2)
=> ([],1)
=> ? = 0 - 2
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,2,3] => [3] => ([],3)
=> ([],1)
=> ? = 0 - 2
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> -1 = 1 - 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> -1 = 1 - 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 2 - 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,2,3,4] => [4] => ([],4)
=> ([],1)
=> ? = 0 - 2
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 1 - 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0 = 2 - 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[1,2,3,4,5] => [5] => ([],5)
=> ([],1)
=> ? = 0 - 2
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 1 - 2
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 1 - 2
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 1 - 2
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 2 - 2
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
[4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
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