- FindStat

Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000638
(load all 16 compositions to match this statistic)

Mapping

St000638: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000452

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000452: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct eigenvalues of a graph.

Matching statistic: St000453

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ 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
[1,2] => [2] => [2] => ([],2)
 => 1
[2,1] => [1,1] => [1,1] => ([(0,1)],2)
 => 2
[1,2,3] => [3] => [3] => ([],3)
 => 1
[1,3,2] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[2,1,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[2,3,1] => [2,1] => [1,2] => ([(1,2)],3)
 => 2
[3,1,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [4] => [4] => ([],4)
 => 1
[1,2,4,3] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[1,4,2,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,4,3] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,3,4,1] => [3,1] => [1,3] => ([(2,3)],4)
 => 2
[2,4,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[2,4,3,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,1,4,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,2,1,4] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,4,1,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 3
[3,4,2,1] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2,3,1] => [1,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,3,1,2] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[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
[1,2,3,4,5] => [5] => [5] => ([],5)
 => 1
[1,2,3,5,4] => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[1,2,4,3,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
[1,2,4,5,3] => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[1,2,5,3,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,3,2,5,4] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,3,4,2,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => [4,1] => [1,4] => ([(3,4)],5)
 => 2
[1,3,5,2,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3
[1,3,5,4,2] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,5,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5] => [2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 3

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000340
(load all 2 compositions to match this statistic)

Mapping

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%

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

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%

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] => [3,1,2] => [1,2] => 110 => 1 = 2 - 1
[2,1,3] => [2,3,1] => [2,1] => 101 => 2 = 3 - 1
[2,3,1] => [2,1,3] => [1,2] => 110 => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,1] => 101 => 2 = 3 - 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] => [4,3,1,2] => [1,1,2] => 1110 => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 2 = 3 - 1
[1,3,4,2] => [4,2,1,3] => [1,1,2] => 1110 => 1 = 2 - 1
[1,4,2,3] => [4,1,3,2] => [1,2,1] => 1101 => 2 = 3 - 1
[1,4,3,2] => [4,1,2,3] => [1,3] => 1100 => 1 = 2 - 1
[2,1,3,4] => [3,4,2,1] => [2,1,1] => 1011 => 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 3 = 4 - 1
[2,3,1,4] => [3,2,4,1] => [1,2,1] => 1101 => 2 = 3 - 1
[2,3,4,1] => [3,2,1,4] => [1,1,2] => 1110 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 2 = 3 - 1
[2,4,3,1] => [3,1,2,4] => [1,3] => 1100 => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [2,1,1] => 1011 => 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 3 = 4 - 1
[3,2,1,4] => [2,3,4,1] => [3,1] => 1001 => 2 = 3 - 1
[3,2,4,1] => [2,3,1,4] => [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] => [2,1,3,4] => [1,3] => 1100 => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => [2,2] => 1010 => 3 = 4 - 1
[4,2,1,3] => [1,3,4,2] => [3,1] => 1001 => 2 = 3 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 3 = 4 - 1
[4,3,1,2] => [1,2,4,3] => [3,1] => 1001 => 2 = 3 - 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] => [5,4,3,1,2] => [1,1,1,2] => 11110 => 1 = 2 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 2 = 3 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 11110 => 1 = 2 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 11101 => 2 = 3 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 11100 => 1 = 2 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 11010 => 3 = 4 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 2 = 3 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 11110 => 1 = 2 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 11101 => 2 = 3 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => 11100 => 1 = 2 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 2 = 3 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 11010 => 3 = 4 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 2 = 3 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => 11010 => 3 = 4 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 11101 => 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: St000777
(load all 3 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 67%

Values

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

Description

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

Matching statistic: St000455

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.