- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000453
(load all 2 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ 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
[2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [4] => ([],4)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000340

Mapping

Mp00064: Permutations —reverse⟶ 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%

Values

[1] => [1] => [1] => [1,0]
 => 0 = 1 - 1
[1,2] => [2,1] => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
[2,1] => [1,2] => [2] => [1,1,0,0]
 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 3 = 4 - 1
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 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 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 5 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 3 - 1

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000691

Mapping

Mp00064: Permutations —reverse⟶ 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%

Values

[1] => [1] => [1] => 1 => 0 = 1 - 1
[1,2] => [2,1] => [1,1] => 11 => 0 = 1 - 1
[2,1] => [1,2] => [2] => 10 => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[1,3,2] => [2,3,1] => [2,1] => 101 => 2 = 3 - 1
[2,1,3] => [3,1,2] => [1,2] => 110 => 1 = 2 - 1
[2,3,1] => [1,3,2] => [2,1] => 101 => 2 = 3 - 1
[3,1,2] => [2,1,3] => [1,2] => 110 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [3] => 100 => 1 = 2 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => 1011 => 2 = 3 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 2 = 3 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => 1101 => 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => 1001 => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => 1110 => 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 3 = 4 - 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => 1011 => 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => 1001 => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => 1110 => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 3 = 4 - 1
[3,2,1,4] => [4,1,2,3] => [1,3] => 1100 => 1 = 2 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => 1010 => 3 = 4 - 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1101 => 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => 1001 => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => 1110 => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => 1010 => 3 = 4 - 1
[4,2,1,3] => [3,1,2,4] => [1,3] => 1100 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 3 = 4 - 1
[4,3,1,2] => [2,1,3,4] => [1,3] => 1100 => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1 = 2 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 2 = 3 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 10111 => 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => 10011 => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 2 = 3 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 2 = 3 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => 10101 => 4 = 5 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 11011 => 2 = 3 - 1

Description

The number of changes of a binary word. This is the number of indices

$i$ such that

$w_i \neq w_{i+1}$ .

Matching statistic: St001504

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => [1,0]
 => 2 = 1 + 1
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 4 = 3 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 5 = 4 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4 = 3 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 6 = 5 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1

Description

The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000796

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => [1] => ? = 1 - 1
[1,2] => [2] => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1 = 2 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0 = 1 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 2 = 3 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 2 = 3 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1 = 2 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1 = 2 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2 = 3 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2 = 3 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 2 = 3 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3 = 4 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2 = 3 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 2 = 3 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2 = 3 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 2 = 3 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3 = 4 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3 = 4 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 2 = 3 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 2 = 3 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3 = 4 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3 = 4 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1 = 2 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 1 = 2 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 2 = 3 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 2 = 3 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 2 = 3 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 2 = 3 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 4 = 5 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 2 = 3 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 2 = 3 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 2 = 3 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 2 = 3 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 4 = 5 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 4 = 5 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 2 = 3 - 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 2 = 3 - 1
[2,1,3,4,5,6,7] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ? = 2 - 1
[2,1,3,4,5,7,6] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 4 - 1
[2,1,3,4,6,5,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,3,4,6,7,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 4 - 1
[2,1,3,4,7,5,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,3,4,7,6,5] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => ? = 4 - 1
[2,1,3,5,4,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 4 - 1
[2,1,3,5,4,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,3,5,6,4,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,3,5,6,7,4] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 4 - 1
[2,1,3,5,7,4,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,3,5,7,6,4] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => ? = 4 - 1
[2,1,3,6,4,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 4 - 1
[2,1,3,6,4,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,3,6,5,4,7] => [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] => ? = 4 - 1
[2,1,3,6,5,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,3,6,7,4,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,3,6,7,5,4] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => ? = 4 - 1
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 4 - 1
[2,1,3,7,4,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,3,7,5,4,6] => [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] => ? = 4 - 1
[2,1,3,7,5,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,3,7,6,4,5] => [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] => ? = 4 - 1
[2,1,3,7,6,5,4] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 4 - 1
[2,1,4,3,5,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 4 - 1
[2,1,4,3,5,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 6 - 1
[2,1,4,3,6,5,7] => [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] => ? = 6 - 1
[2,1,4,3,6,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 6 - 1
[2,1,4,3,7,5,6] => [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] => ? = 6 - 1
[2,1,4,3,7,6,5] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 6 - 1
[2,1,4,5,3,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 4 - 1
[2,1,4,5,3,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,4,5,6,3,7] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,4,5,6,7,3] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => ? = 4 - 1
[2,1,4,5,7,3,6] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,4,5,7,6,3] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => ? = 4 - 1
[2,1,4,6,3,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 4 - 1
[2,1,4,6,3,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,4,6,5,3,7] => [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] => ? = 4 - 1
[2,1,4,6,5,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,4,6,7,3,5] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => ? = 4 - 1
[2,1,4,6,7,5,3] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => ? = 4 - 1
[2,1,4,7,3,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 4 - 1
[2,1,4,7,3,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,4,7,5,3,6] => [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] => ? = 4 - 1
[2,1,4,7,5,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 6 - 1
[2,1,4,7,6,3,5] => [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] => ? = 4 - 1
[2,1,4,7,6,5,3] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 4 - 1
[2,1,5,3,4,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 4 - 1

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: St000777

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 1
[1,2] => [2] => ([],2)
 => ? = 1
[2,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,3] => [3] => ([],3)
 => ? = 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [4] => ([],4)
 => ? = 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => ? = 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[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)
 => 3
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ? = 2
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[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)
 => 4
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[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)
 => 3
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ? = 2
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[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)
 => 4
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 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)
 => 4
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[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)
 => 4
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 3

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Matching statistic: St000638
(load all 11 compositions to match this statistic)

Mapping

Mp00126: Permutations —cactus evacuation⟶ Permutations
St000638: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 86%

Values

[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [3,1,2] => 3
[2,1,3] => [2,3,1] => 2
[2,3,1] => [2,1,3] => 3
[3,1,2] => [1,3,2] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => 3
[1,3,4,2] => [3,1,2,4] => 3
[1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [4,3,1,2] => 3
[2,1,3,4] => [2,3,4,1] => 2
[2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [2,3,1,4] => 3
[2,3,4,1] => [2,1,3,4] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [4,2,1,3] => 3
[3,1,2,4] => [1,3,4,2] => 2
[3,1,4,2] => [3,1,4,2] => 4
[3,2,1,4] => [3,4,2,1] => 2
[3,2,4,1] => [3,2,4,1] => 4
[3,4,1,2] => [3,4,1,2] => 3
[3,4,2,1] => [3,2,1,4] => 3
[4,1,2,3] => [1,2,4,3] => 2
[4,1,3,2] => [4,1,3,2] => 4
[4,2,1,3] => [2,4,3,1] => 2
[4,2,3,1] => [4,2,3,1] => 4
[4,3,1,2] => [1,4,3,2] => 2
[4,3,2,1] => [4,3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [5,1,2,3,4] => 3
[1,2,4,3,5] => [1,4,2,3,5] => 3
[1,2,4,5,3] => [4,1,2,3,5] => 3
[1,2,5,3,4] => [1,5,2,3,4] => 3
[1,2,5,4,3] => [5,4,1,2,3] => 3
[1,3,2,4,5] => [1,3,4,2,5] => 3
[1,3,2,5,4] => [3,1,5,2,4] => 5
[1,3,4,2,5] => [1,3,2,4,5] => 3
[1,3,4,5,2] => [3,1,2,4,5] => 3
[1,3,5,2,4] => [3,5,1,2,4] => 3
[1,3,5,4,2] => [5,3,1,2,4] => 3
[1,4,2,3,5] => [1,2,4,3,5] => 3
[1,4,2,5,3] => [4,1,5,2,3] => 5
[1,4,3,2,5] => [1,4,3,2,5] => 3
[1,4,3,5,2] => [4,1,3,2,5] => 5
[1,4,5,2,3] => [4,5,1,2,3] => 3
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 1
[1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 3
[1,2,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 3
[1,2,3,4,6,7,5] => [6,1,2,3,4,5,7] => ? = 3
[1,2,3,4,7,5,6] => [1,7,2,3,4,5,6] => ? = 3
[1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 3
[1,2,3,5,4,6,7] => [1,2,5,3,4,6,7] => ? = 3
[1,2,3,5,4,7,6] => [5,1,7,2,3,4,6] => ? = 5
[1,2,3,5,6,4,7] => [1,5,2,3,4,6,7] => ? = 3
[1,2,3,5,6,7,4] => [5,1,2,3,4,6,7] => ? = 3
[1,2,3,5,7,4,6] => [5,7,1,2,3,4,6] => ? = 3
[1,2,3,5,7,6,4] => [7,5,1,2,3,4,6] => ? = 3
[1,2,3,6,4,5,7] => [1,2,6,3,4,5,7] => ? = 3
[1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 5
[1,2,3,6,5,4,7] => [1,6,5,2,3,4,7] => ? = 3
[1,2,3,6,5,7,4] => [6,1,5,2,3,4,7] => ? = 5
[1,2,3,6,7,4,5] => [6,7,1,2,3,4,5] => ? = 3
[1,2,3,6,7,5,4] => [6,5,1,2,3,4,7] => ? = 3
[1,2,3,7,4,5,6] => [1,2,7,3,4,5,6] => ? = 3
[1,2,3,7,4,6,5] => [7,1,6,2,3,4,5] => ? = 5
[1,2,3,7,5,4,6] => [1,7,5,2,3,4,6] => ? = 3
[1,2,3,7,5,6,4] => [7,1,5,2,3,4,6] => ? = 5
[1,2,3,7,6,4,5] => [1,7,6,2,3,4,5] => ? = 3
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3
[1,2,4,3,5,6,7] => [1,2,4,5,3,6,7] => ? = 3
[1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 5
[1,2,4,3,6,5,7] => [1,4,2,6,3,5,7] => ? = 5
[1,2,4,3,6,7,5] => [4,1,2,6,3,5,7] => ? = 5
[1,2,4,3,7,5,6] => [1,4,2,7,3,5,6] => ? = 5
[1,2,4,3,7,6,5] => [7,4,1,6,2,3,5] => ? = 5
[1,2,4,5,3,6,7] => [1,2,4,3,5,6,7] => ? = 3
[1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => ? = 5
[1,2,4,5,6,3,7] => [1,4,2,3,5,6,7] => ? = 3
[1,2,4,5,6,7,3] => [4,1,2,3,5,6,7] => ? = 3
[1,2,4,5,7,3,6] => [4,7,1,2,3,5,6] => ? = 3
[1,2,4,5,7,6,3] => [7,4,1,2,3,5,6] => ? = 3
[1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => ? = 3
[1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => ? = 5
[1,2,4,6,5,3,7] => [1,6,4,2,3,5,7] => ? = 3
[1,2,4,6,5,7,3] => [6,1,4,2,3,5,7] => ? = 5
[1,2,4,6,7,3,5] => [4,6,1,2,3,5,7] => ? = 3
[1,2,4,6,7,5,3] => [6,4,1,2,3,5,7] => ? = 3
[1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => ? = 3
[1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => ? = 5
[1,2,4,7,5,3,6] => [1,7,4,2,3,5,6] => ? = 3
[1,2,4,7,5,6,3] => [7,1,4,2,3,5,6] => ? = 5
[1,2,4,7,6,3,5] => [4,7,6,1,2,3,5] => ? = 3
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 3
[1,2,5,3,4,6,7] => [1,2,3,5,4,6,7] => ? = 3
[1,2,5,3,4,7,6] => [5,1,2,7,3,4,6] => ? = 5

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: St000388

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St000388: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 86%

Values

[1] => [1] => ([],1)
 => ([],2)
 => 1
[1,2] => [2] => ([],2)
 => ([],3)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(1,2)],3)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],4)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4] => [4] => ([],4)
 => ([],5)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => ([],6)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,3,4,5,6,7] => [7] => ([],7)
 => ([],8)
 => ? = 1
[1,2,3,4,5,7,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,6,5,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,6,7,5] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,7,5,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,7,6,5] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,4,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,5,6,4,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,6,7,4] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,7,4,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,7,6,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,4,7,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,6,5,4,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,5,7,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,6,7,4,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,7,5,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,4,6,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,7,5,4,6] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,5,6,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,7,6,4,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,6,5,4] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
[1,2,4,3,5,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,3,5,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,6,5,7] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,6,7,5] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,7,5,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,7,6,5] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 5
[1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,3,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,5,6,3,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,6,7,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,7,3,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,7,6,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,3,7,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,6,5,3,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,5,7,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,6,7,3,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,7,5,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,3,6,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,7,5,3,6] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,5,6,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,7,6,3,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,6,5,3] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
[1,2,5,3,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,5,3,4,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(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

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St001951: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 86%

Values

[1] => [1] => ([],1)
 => ([],2)
 => 1
[1,2] => [2] => ([],2)
 => ([],3)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(1,2)],3)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],4)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4] => [4] => ([],4)
 => ([],5)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => ([],6)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,3,4,5,6,7] => [7] => ([],7)
 => ([],8)
 => ? = 1
[1,2,3,4,5,7,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,6,5,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,6,7,5] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,7,5,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,7,6,5] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,4,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,5,6,4,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,6,7,4] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,7,4,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,7,6,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,4,7,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,6,5,4,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,5,7,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,6,7,4,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,7,5,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,4,6,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,7,5,4,6] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,5,6,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,7,6,4,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,6,5,4] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
[1,2,4,3,5,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,3,5,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,6,5,7] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,6,7,5] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,7,5,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,7,6,5] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 5
[1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,3,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,5,6,3,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,6,7,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,7,3,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,7,6,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,3,7,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,6,5,3,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,5,7,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,6,7,3,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,7,5,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,3,6,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,7,5,3,6] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,5,6,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,7,6,3,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,6,5,3] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
[1,2,5,3,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,5,3,4,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(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: St001352

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St001352: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 86%

Values

[1] => [1] => ([],1)
 => ([],2)
 => 1
[1,2] => [2] => ([],2)
 => ([],3)
 => 1
[2,1] => [1,1] => ([(0,1)],2)
 => ([(1,2)],3)
 => 2
[1,2,3] => [3] => ([],3)
 => ([],4)
 => 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 3
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 3
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4] => [4] => ([],4)
 => ([],5)
 => 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 3
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => ([],6)
 => 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 3
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
[6,5,4,3,2,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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
[1,2,3,4,5,6,7] => [7] => ([],7)
 => ([],8)
 => ? = 1
[1,2,3,4,5,7,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,6,5,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,6,7,5] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,7,5,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,4,7,6,5] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,4,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,4,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,5,6,4,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,6,7,4] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,7,4,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,5,7,6,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,4,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,4,7,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,6,5,4,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,5,7,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,6,7,4,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,6,7,5,4] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,4,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,4,6,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,7,5,4,6] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,5,6,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,3,7,6,4,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,3,7,6,5,4] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
[1,2,4,3,5,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,3,5,7,6] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,6,5,7] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,6,7,5] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,7,5,6] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,3,7,6,5] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 5
[1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,3,7,6] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,5,6,3,7] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,6,7,3] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,7,3,6] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,5,7,6,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,3,7,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,6,5,3,7] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,5,7,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,6,7,3,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,6,7,5,3] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,3,6,5] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,7,5,3,6] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,5,6,3] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[1,2,4,7,6,3,5] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[1,2,4,7,6,5,3] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
[1,2,5,3,4,6,7] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(4,7),(5,7),(6,7)],8)
 => ? = 3

Description

The number of internal nodes in the modular decomposition of a graph.

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St001877Number of indecomposable injective modules with projective dimension 2.