- FindStat

Your data matches 6 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

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001035

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001035: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 100%●distinct values known / distinct values provided: 86%

Values

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

Description

The convexity degree of the parallelogram polyomino associated with the Dyck path. A parallelogram polyomino is

$k$ -convex if

$k$ is the maximal number of turns an axis-parallel path must take to connect two cells of the polyomino. For example, any rotation of a Ferrers shape has convexity degree at most one. The (bivariate) generating function is given in Theorem 2 of [1].

Matching statistic: St000453

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 66%●distinct values known / distinct values provided: 29%

Values

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

Matching statistic: St001488

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 71%

Values

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

Description

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