- FindStat

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001486
(load all 4 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.

Matching statistic: St000777

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000453

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of distinct Laplacian eigenvalues of a graph.

Matching statistic: St000455

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 17%

Values

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

Description

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

Matching statistic: St001488

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.