- FindStat

Your data matches 28 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000264
(load all 3 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00264: Graphs —delete endpoints⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St000455
(load all 6 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 46%●distinct values known / distinct values provided: 33%

Values

([],3)
 => ([],3)
 => [3] => ([],3)
 => ? = 3 - 3
([],4)
 => ([],4)
 => [4] => ([],4)
 => ? = 3 - 3
([(2,3)],4)
 => ([(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 3 - 3
([],5)
 => ([],5)
 => [5] => ([],5)
 => ? = 3 - 3
([(3,4)],5)
 => ([(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
([(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 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)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 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)
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 3 - 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)
 => [2,2,1] => ([(0,4),(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
([],6)
 => ([],6)
 => [6] => ([],6)
 => ? = 3 - 3
([(4,5)],6)
 => ([(4,5)],6)
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
 => ([(1,2),(3,5),(4,5)],6)
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => [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
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(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)
 => [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
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [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
([(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)
 => [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
([(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)
 => [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
([(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)
 => [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
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 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)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 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)
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 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)
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [2,2,2] => ([(1,5),(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)
 => [2,2,2] => ([(1,5),(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)
 => [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)
 => ? = 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)
 => [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)
 => ? = 4 - 3
([],7)
 => ([],7)
 => [7] => ([],7)
 => ? = 3 - 3
([(5,6)],7)
 => ([(5,6)],7)
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(4,6),(5,6)],7)
 => ([(4,6),(5,6)],7)
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
 => ([(3,6),(4,6),(5,6)],7)
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(3,6),(4,5)],7)
 => ([(3,6),(4,5)],7)
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
 => ([(2,3),(4,6),(5,6)],7)
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
 => ([(4,5),(4,6),(5,6)],7)
 => [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
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(1,2),(3,6),(4,6),(5,6)],7)
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(1,6),(2,6),(3,5),(4,5)],7)
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [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
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 3
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 3
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 3
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 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: St000781

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

Matching statistic: St001901

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.

Matching statistic: St001934

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of monotone factorisations of genus zero of a permutation of given cycle type. A monotone factorisation of genus zero of a permutation

$\pi\in\mathfrak S_n$ with

$\ell$ cycles, including fixed points, is a tuple of

$r = n - \ell$ transpositions

$(a_1, b_1),\dots,(a_r, b_r)$ with

$b_1 \leq \dots \leq b_r$ and

$a_i < b_i$ for all

$i$ , whose product, in this order, is

$\pi$ . For example, the cycle

$(2,3,1)$ has the two factorizations

$(2,3)(1,3)$ and

$(1,2)(2,3)$ .

Matching statistic: St000205

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given

$\lambda$ count how many ''integer compositions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St001175

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

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

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001719
(load all 14 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

([],3)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([],4)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(2,3)],4)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([],5)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 5 - 2
([],6)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,16),(1,21),(2,10),(2,11),(2,12),(2,16),(2,20),(3,8),(3,9),(3,12),(3,15),(3,19),(4,7),(4,9),(4,11),(4,14),(4,18),(5,7),(5,8),(5,10),(5,13),(5,17),(6,17),(6,18),(6,19),(6,20),(6,21),(7,22),(7,25),(7,28),(7,34),(8,22),(8,23),(8,26),(8,32),(9,22),(9,24),(9,27),(9,33),(10,23),(10,25),(10,29),(10,35),(11,24),(11,25),(11,30),(11,36),(12,23),(12,24),(12,31),(12,37),(13,26),(13,28),(13,29),(13,38),(14,27),(14,28),(14,30),(14,39),(15,26),(15,27),(15,31),(15,40),(16,29),(16,30),(16,31),(16,41),(17,32),(17,34),(17,35),(17,38),(18,33),(18,34),(18,36),(18,39),(19,32),(19,33),(19,37),(19,40),(20,35),(20,36),(20,37),(20,41),(21,38),(21,39),(21,40),(21,41),(22,51),(22,52),(23,48),(23,52),(24,49),(24,52),(25,50),(25,52),(26,42),(26,52),(27,43),(27,52),(28,44),(28,52),(29,45),(29,52),(30,46),(30,52),(31,47),(31,52),(32,42),(32,48),(32,51),(33,43),(33,49),(33,51),(34,44),(34,50),(34,51),(35,45),(35,48),(35,50),(36,46),(36,49),(36,50),(37,47),(37,48),(37,49),(38,42),(38,44),(38,45),(39,43),(39,44),(39,46),(40,42),(40,43),(40,47),(41,45),(41,46),(41,47),(42,53),(43,53),(44,53),(45,53),(46,53),(47,53),(48,53),(49,53),(50,53),(51,53),(52,53)],54)
 => ? = 5 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 3 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3 - 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 3 - 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
 => ? = 3 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 3 - 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 3 - 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(1,18),(1,19),(1,20),(1,21),(2,13),(2,14),(2,15),(2,16),(2,21),(3,10),(3,11),(3,12),(3,16),(3,20),(4,8),(4,9),(4,12),(4,15),(4,19),(5,7),(5,9),(5,11),(5,14),(5,18),(6,7),(6,8),(6,10),(6,13),(6,17),(7,22),(7,25),(7,28),(7,34),(8,22),(8,23),(8,26),(8,32),(9,22),(9,24),(9,27),(9,33),(10,23),(10,25),(10,29),(10,35),(11,24),(11,25),(11,30),(11,36),(12,23),(12,24),(12,31),(12,37),(13,26),(13,28),(13,29),(13,38),(14,27),(14,28),(14,30),(14,39),(15,26),(15,27),(15,31),(15,40),(16,29),(16,30),(16,31),(16,41),(17,32),(17,34),(17,35),(17,38),(18,33),(18,34),(18,36),(18,39),(19,32),(19,33),(19,37),(19,40),(20,35),(20,36),(20,37),(20,41),(21,38),(21,39),(21,40),(21,41),(22,45),(22,46),(22,56),(23,42),(23,46),(23,53),(24,43),(24,46),(24,54),(25,44),(25,46),(25,55),(26,42),(26,45),(26,47),(27,43),(27,45),(27,48),(28,44),(28,45),(28,49),(29,42),(29,44),(29,50),(30,43),(30,44),(30,51),(31,42),(31,43),(31,52),(32,47),(32,53),(32,56),(33,48),(33,54),(33,56),(34,49),(34,55),(34,56),(35,50),(35,53),(35,55),(36,51),(36,54),(36,55),(37,52),(37,53),(37,54),(38,47),(38,49),(38,50),(39,48),(39,49),(39,51),(40,47),(40,48),(40,52),(41,50),(41,51),(41,52),(42,57),(43,57),(44,57),(45,57),(46,57),(47,57),(48,57),(49,57),(50,57),(51,57),(52,57),(53,57),(54,57),(55,57),(56,57)],58)
 => ? = 3 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,16),(1,26),(1,27),(1,28),(2,8),(2,10),(2,15),(2,23),(2,24),(2,25),(3,8),(3,9),(3,14),(3,20),(3,21),(3,22),(4,12),(4,13),(4,19),(4,22),(4,25),(4,28),(5,11),(5,13),(5,18),(5,21),(5,24),(5,27),(6,11),(6,12),(6,17),(6,20),(6,23),(6,26),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(8,35),(8,36),(8,37),(8,66),(9,29),(9,30),(9,31),(9,66),(10,32),(10,33),(10,34),(10,66),(11,40),(11,43),(11,46),(11,65),(12,38),(12,41),(12,44),(12,65),(13,39),(13,42),(13,45),(13,65),(14,47),(14,48),(14,49),(14,66),(15,50),(15,51),(15,52),(15,66),(16,53),(16,54),(16,55),(16,66),(17,47),(17,50),(17,53),(17,65),(18,48),(18,51),(18,54),(18,65),(19,49),(19,52),(19,55),(19,65),(20,29),(20,35),(20,38),(20,40),(20,47),(21,30),(21,36),(21,39),(21,40),(21,48),(22,31),(22,37),(22,38),(22,39),(22,49),(23,32),(23,35),(23,41),(23,43),(23,50),(24,33),(24,36),(24,42),(24,43),(24,51),(25,34),(25,37),(25,41),(25,42),(25,52),(26,29),(26,32),(26,44),(26,46),(26,53),(27,30),(27,33),(27,45),(27,46),(27,54),(28,31),(28,34),(28,44),(28,45),(28,55),(29,56),(29,58),(29,67),(30,57),(30,58),(30,68),(31,56),(31,57),(31,69),(32,59),(32,61),(32,67),(33,60),(33,61),(33,68),(34,59),(34,60),(34,69),(35,62),(35,64),(35,67),(36,63),(36,64),(36,68),(37,62),(37,63),(37,69),(38,56),(38,62),(38,70),(39,57),(39,63),(39,70),(40,58),(40,64),(40,70),(41,59),(41,62),(41,71),(42,60),(42,63),(42,71),(43,61),(43,64),(43,71),(44,56),(44,59),(44,72),(45,57),(45,60),(45,72),(46,58),(46,61),(46,72),(47,67),(47,70),(48,68),(48,70),(49,69),(49,70),(50,67),(50,71),(51,68),(51,71),(52,69),(52,71),(53,67),(53,72),(54,68),(54,72),(55,69),(55,72),(56,73),(57,73),(58,73),(59,73),(60,73),(61,73),(62,73),(63,73),(64,73),(65,70),(65,71),(65,72),(66,67),(66,68),(66,69),(67,73),(68,73),(69,73),(70,73),(71,73),(72,73)],74)
 => ? = 3 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,12),(5,13),(5,14),(5,15),(5,26),(6,8),(6,9),(6,10),(6,11),(6,26),(7,16),(7,17),(7,18),(7,19),(7,26),(8,37),(8,38),(8,40),(8,57),(9,37),(9,39),(9,41),(9,58),(10,38),(10,39),(10,42),(10,59),(11,40),(11,41),(11,42),(11,60),(12,43),(12,44),(12,46),(12,57),(13,43),(13,45),(13,47),(13,58),(14,44),(14,45),(14,48),(14,59),(15,46),(15,47),(15,48),(15,60),(16,31),(16,32),(16,34),(16,57),(17,31),(17,33),(17,35),(17,58),(18,32),(18,33),(18,36),(18,59),(19,34),(19,35),(19,36),(19,60),(20,27),(20,30),(20,31),(20,37),(20,43),(21,27),(21,28),(21,32),(21,38),(21,44),(22,27),(22,29),(22,33),(22,39),(22,45),(23,28),(23,30),(23,34),(23,40),(23,46),(24,29),(24,30),(24,35),(24,41),(24,47),(25,28),(25,29),(25,36),(25,42),(25,48),(26,57),(26,58),(26,59),(26,60),(27,49),(27,53),(27,67),(28,50),(28,54),(28,67),(29,51),(29,55),(29,67),(30,52),(30,56),(30,67),(31,63),(31,67),(32,61),(32,67),(33,62),(33,67),(34,64),(34,67),(35,65),(35,67),(36,66),(36,67),(37,49),(37,52),(37,63),(38,49),(38,50),(38,61),(39,49),(39,51),(39,62),(40,50),(40,52),(40,64),(41,51),(41,52),(41,65),(42,50),(42,51),(42,66),(43,53),(43,56),(43,63),(44,53),(44,54),(44,61),(45,53),(45,55),(45,62),(46,54),(46,56),(46,64),(47,55),(47,56),(47,65),(48,54),(48,55),(48,66),(49,68),(50,68),(51,68),(52,68),(53,68),(54,68),(55,68),(56,68),(57,61),(57,63),(57,64),(58,62),(58,63),(58,65),(59,61),(59,62),(59,66),(60,64),(60,65),(60,66),(61,68),(62,68),(63,68),(64,68),(65,68),(66,68),(67,68)],69)
 => ? = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 4 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(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),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,10),(1,11),(1,40),(1,41),(2,15),(2,16),(2,17),(2,26),(2,27),(2,41),(3,12),(3,13),(3,14),(3,24),(3,25),(3,41),(4,21),(4,22),(4,23),(4,25),(4,27),(4,40),(5,18),(5,19),(5,20),(5,24),(5,26),(5,40),(6,11),(6,14),(6,17),(6,20),(6,23),(6,42),(7,10),(7,13),(7,16),(7,19),(7,22),(7,42),(8,9),(8,12),(8,15),(8,18),(8,21),(8,42),(9,43),(9,46),(9,49),(10,44),(10,47),(10,49),(11,45),(11,48),(11,49),(12,28),(12,31),(12,43),(12,50),(13,29),(13,32),(13,44),(13,50),(14,30),(14,33),(14,45),(14,50),(15,34),(15,37),(15,43),(15,51),(16,35),(16,38),(16,44),(16,51),(17,36),(17,39),(17,45),(17,51),(18,28),(18,34),(18,46),(18,52),(19,29),(19,35),(19,47),(19,52),(20,30),(20,36),(20,48),(20,52),(21,31),(21,37),(21,46),(21,53),(22,32),(22,38),(22,47),(22,53),(23,33),(23,39),(23,48),(23,53),(24,28),(24,29),(24,30),(24,58),(25,31),(25,32),(25,33),(25,58),(26,34),(26,35),(26,36),(26,58),(27,37),(27,38),(27,39),(27,58),(28,54),(28,61),(29,54),(29,62),(30,54),(30,63),(31,55),(31,61),(32,55),(32,62),(33,55),(33,63),(34,56),(34,61),(35,56),(35,62),(36,56),(36,63),(37,57),(37,61),(38,57),(38,62),(39,57),(39,63),(40,46),(40,47),(40,48),(40,58),(41,43),(41,44),(41,45),(41,58),(42,49),(42,50),(42,51),(42,52),(42,53),(43,59),(43,61),(44,59),(44,62),(45,59),(45,63),(46,60),(46,61),(47,60),(47,62),(48,60),(48,63),(49,59),(49,60),(50,54),(50,55),(50,59),(51,56),(51,57),(51,59),(52,54),(52,56),(52,60),(53,55),(53,57),(53,60),(54,64),(55,64),(56,64),(57,64),(58,61),(58,62),(58,63),(59,64),(60,64),(61,64),(62,64),(63,64)],65)
 => ? = 4 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,17),(1,23),(1,30),(1,42),(1,44),(2,10),(2,16),(2,22),(2,30),(2,41),(2,43),(3,12),(3,18),(3,24),(3,29),(3,41),(3,44),(4,13),(4,19),(4,25),(4,29),(4,42),(4,43),(5,15),(5,21),(5,27),(5,28),(5,43),(5,44),(6,14),(6,20),(6,26),(6,28),(6,41),(6,42),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(7,31),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,31),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,31),(10,32),(10,45),(10,47),(10,57),(11,32),(11,46),(11,48),(11,58),(12,33),(12,45),(12,48),(12,59),(13,33),(13,46),(13,47),(13,60),(14,34),(14,45),(14,46),(14,61),(15,34),(15,47),(15,48),(15,62),(16,35),(16,49),(16,51),(16,57),(17,35),(17,50),(17,52),(17,58),(18,36),(18,49),(18,52),(18,59),(19,36),(19,50),(19,51),(19,60),(20,37),(20,49),(20,50),(20,61),(21,37),(21,51),(21,52),(21,62),(22,38),(22,53),(22,55),(22,57),(23,38),(23,54),(23,56),(23,58),(24,39),(24,53),(24,56),(24,59),(25,39),(25,54),(25,55),(25,60),(26,40),(26,53),(26,54),(26,61),(27,40),(27,55),(27,56),(27,62),(28,34),(28,37),(28,40),(28,70),(29,33),(29,36),(29,39),(29,70),(30,32),(30,35),(30,38),(30,70),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(32,63),(32,71),(33,64),(33,71),(34,65),(34,71),(35,63),(35,72),(36,64),(36,72),(37,65),(37,72),(38,63),(38,73),(39,64),(39,73),(40,65),(40,73),(41,45),(41,49),(41,53),(41,70),(42,46),(42,50),(42,54),(42,70),(43,47),(43,51),(43,55),(43,70),(44,48),(44,52),(44,56),(44,70),(45,66),(45,71),(46,67),(46,71),(47,68),(47,71),(48,69),(48,71),(49,66),(49,72),(50,67),(50,72),(51,68),(51,72),(52,69),(52,72),(53,66),(53,73),(54,67),(54,73),(55,68),(55,73),(56,69),(56,73),(57,63),(57,66),(57,68),(58,63),(58,67),(58,69),(59,64),(59,66),(59,69),(60,64),(60,67),(60,68),(61,65),(61,66),(61,67),(62,65),(62,68),(62,69),(63,74),(64,74),(65,74),(66,74),(67,74),(68,74),(69,74),(70,71),(70,72),(70,73),(71,74),(72,74),(73,74)],75)
 => ? = 4 - 2
([],7)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 3 - 2
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 3 - 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 3 - 2
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,11),(1,12),(1,16),(1,20),(2,8),(2,9),(2,12),(2,15),(2,19),(3,7),(3,9),(3,11),(3,14),(3,18),(4,7),(4,8),(4,10),(4,13),(4,17),(5,17),(5,18),(5,19),(5,20),(5,21),(6,13),(6,14),(6,15),(6,16),(6,21),(7,24),(7,30),(7,44),(8,22),(8,28),(8,44),(9,23),(9,29),(9,44),(10,25),(10,31),(10,44),(11,26),(11,32),(11,44),(12,27),(12,33),(12,44),(13,22),(13,24),(13,25),(13,34),(14,23),(14,24),(14,26),(14,35),(15,22),(15,23),(15,27),(15,36),(16,25),(16,26),(16,27),(16,37),(17,28),(17,30),(17,31),(17,34),(18,29),(18,30),(18,32),(18,35),(19,28),(19,29),(19,33),(19,36),(20,31),(20,32),(20,33),(20,37),(21,34),(21,35),(21,36),(21,37),(22,38),(22,45),(23,39),(23,45),(24,40),(24,45),(25,41),(25,45),(26,42),(26,45),(27,43),(27,45),(28,38),(28,46),(29,39),(29,46),(30,40),(30,46),(31,41),(31,46),(32,42),(32,46),(33,43),(33,46),(34,38),(34,40),(34,41),(35,39),(35,40),(35,42),(36,38),(36,39),(36,43),(37,41),(37,42),(37,43),(38,47),(39,47),(40,47),(41,47),(42,47),(43,47),(44,45),(44,46),(45,47),(46,47)],48)
 => ? = 3 - 2

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St001820
(load all 12 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%

Values

([],3)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([],4)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(2,3)],4)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([],5)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 5 - 2
([],6)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(0,5),(1,4),(2,3)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,16),(1,21),(2,10),(2,11),(2,12),(2,16),(2,20),(3,8),(3,9),(3,12),(3,15),(3,19),(4,7),(4,9),(4,11),(4,14),(4,18),(5,7),(5,8),(5,10),(5,13),(5,17),(6,17),(6,18),(6,19),(6,20),(6,21),(7,22),(7,25),(7,28),(7,34),(8,22),(8,23),(8,26),(8,32),(9,22),(9,24),(9,27),(9,33),(10,23),(10,25),(10,29),(10,35),(11,24),(11,25),(11,30),(11,36),(12,23),(12,24),(12,31),(12,37),(13,26),(13,28),(13,29),(13,38),(14,27),(14,28),(14,30),(14,39),(15,26),(15,27),(15,31),(15,40),(16,29),(16,30),(16,31),(16,41),(17,32),(17,34),(17,35),(17,38),(18,33),(18,34),(18,36),(18,39),(19,32),(19,33),(19,37),(19,40),(20,35),(20,36),(20,37),(20,41),(21,38),(21,39),(21,40),(21,41),(22,51),(22,52),(23,48),(23,52),(24,49),(24,52),(25,50),(25,52),(26,42),(26,52),(27,43),(27,52),(28,44),(28,52),(29,45),(29,52),(30,46),(30,52),(31,47),(31,52),(32,42),(32,48),(32,51),(33,43),(33,49),(33,51),(34,44),(34,50),(34,51),(35,45),(35,48),(35,50),(36,46),(36,49),(36,50),(37,47),(37,48),(37,49),(38,42),(38,44),(38,45),(39,43),(39,44),(39,46),(40,42),(40,43),(40,47),(41,45),(41,46),(41,47),(42,53),(43,53),(44,53),(45,53),(46,53),(47,53),(48,53),(49,53),(50,53),(51,53),(52,53)],54)
 => ? = 5 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 3 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3 - 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 3 - 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
 => ? = 3 - 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 3 - 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 3 - 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(1,18),(1,19),(1,20),(1,21),(2,13),(2,14),(2,15),(2,16),(2,21),(3,10),(3,11),(3,12),(3,16),(3,20),(4,8),(4,9),(4,12),(4,15),(4,19),(5,7),(5,9),(5,11),(5,14),(5,18),(6,7),(6,8),(6,10),(6,13),(6,17),(7,22),(7,25),(7,28),(7,34),(8,22),(8,23),(8,26),(8,32),(9,22),(9,24),(9,27),(9,33),(10,23),(10,25),(10,29),(10,35),(11,24),(11,25),(11,30),(11,36),(12,23),(12,24),(12,31),(12,37),(13,26),(13,28),(13,29),(13,38),(14,27),(14,28),(14,30),(14,39),(15,26),(15,27),(15,31),(15,40),(16,29),(16,30),(16,31),(16,41),(17,32),(17,34),(17,35),(17,38),(18,33),(18,34),(18,36),(18,39),(19,32),(19,33),(19,37),(19,40),(20,35),(20,36),(20,37),(20,41),(21,38),(21,39),(21,40),(21,41),(22,45),(22,46),(22,56),(23,42),(23,46),(23,53),(24,43),(24,46),(24,54),(25,44),(25,46),(25,55),(26,42),(26,45),(26,47),(27,43),(27,45),(27,48),(28,44),(28,45),(28,49),(29,42),(29,44),(29,50),(30,43),(30,44),(30,51),(31,42),(31,43),(31,52),(32,47),(32,53),(32,56),(33,48),(33,54),(33,56),(34,49),(34,55),(34,56),(35,50),(35,53),(35,55),(36,51),(36,54),(36,55),(37,52),(37,53),(37,54),(38,47),(38,49),(38,50),(39,48),(39,49),(39,51),(40,47),(40,48),(40,52),(41,50),(41,51),(41,52),(42,57),(43,57),(44,57),(45,57),(46,57),(47,57),(48,57),(49,57),(50,57),(51,57),(52,57),(53,57),(54,57),(55,57),(56,57)],58)
 => ? = 3 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,16),(1,26),(1,27),(1,28),(2,8),(2,10),(2,15),(2,23),(2,24),(2,25),(3,8),(3,9),(3,14),(3,20),(3,21),(3,22),(4,12),(4,13),(4,19),(4,22),(4,25),(4,28),(5,11),(5,13),(5,18),(5,21),(5,24),(5,27),(6,11),(6,12),(6,17),(6,20),(6,23),(6,26),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(8,35),(8,36),(8,37),(8,66),(9,29),(9,30),(9,31),(9,66),(10,32),(10,33),(10,34),(10,66),(11,40),(11,43),(11,46),(11,65),(12,38),(12,41),(12,44),(12,65),(13,39),(13,42),(13,45),(13,65),(14,47),(14,48),(14,49),(14,66),(15,50),(15,51),(15,52),(15,66),(16,53),(16,54),(16,55),(16,66),(17,47),(17,50),(17,53),(17,65),(18,48),(18,51),(18,54),(18,65),(19,49),(19,52),(19,55),(19,65),(20,29),(20,35),(20,38),(20,40),(20,47),(21,30),(21,36),(21,39),(21,40),(21,48),(22,31),(22,37),(22,38),(22,39),(22,49),(23,32),(23,35),(23,41),(23,43),(23,50),(24,33),(24,36),(24,42),(24,43),(24,51),(25,34),(25,37),(25,41),(25,42),(25,52),(26,29),(26,32),(26,44),(26,46),(26,53),(27,30),(27,33),(27,45),(27,46),(27,54),(28,31),(28,34),(28,44),(28,45),(28,55),(29,56),(29,58),(29,67),(30,57),(30,58),(30,68),(31,56),(31,57),(31,69),(32,59),(32,61),(32,67),(33,60),(33,61),(33,68),(34,59),(34,60),(34,69),(35,62),(35,64),(35,67),(36,63),(36,64),(36,68),(37,62),(37,63),(37,69),(38,56),(38,62),(38,70),(39,57),(39,63),(39,70),(40,58),(40,64),(40,70),(41,59),(41,62),(41,71),(42,60),(42,63),(42,71),(43,61),(43,64),(43,71),(44,56),(44,59),(44,72),(45,57),(45,60),(45,72),(46,58),(46,61),(46,72),(47,67),(47,70),(48,68),(48,70),(49,69),(49,70),(50,67),(50,71),(51,68),(51,71),(52,69),(52,71),(53,67),(53,72),(54,68),(54,72),(55,69),(55,72),(56,73),(57,73),(58,73),(59,73),(60,73),(61,73),(62,73),(63,73),(64,73),(65,70),(65,71),(65,72),(66,67),(66,68),(66,69),(67,73),(68,73),(69,73),(70,73),(71,73),(72,73)],74)
 => ? = 3 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,12),(5,13),(5,14),(5,15),(5,26),(6,8),(6,9),(6,10),(6,11),(6,26),(7,16),(7,17),(7,18),(7,19),(7,26),(8,37),(8,38),(8,40),(8,57),(9,37),(9,39),(9,41),(9,58),(10,38),(10,39),(10,42),(10,59),(11,40),(11,41),(11,42),(11,60),(12,43),(12,44),(12,46),(12,57),(13,43),(13,45),(13,47),(13,58),(14,44),(14,45),(14,48),(14,59),(15,46),(15,47),(15,48),(15,60),(16,31),(16,32),(16,34),(16,57),(17,31),(17,33),(17,35),(17,58),(18,32),(18,33),(18,36),(18,59),(19,34),(19,35),(19,36),(19,60),(20,27),(20,30),(20,31),(20,37),(20,43),(21,27),(21,28),(21,32),(21,38),(21,44),(22,27),(22,29),(22,33),(22,39),(22,45),(23,28),(23,30),(23,34),(23,40),(23,46),(24,29),(24,30),(24,35),(24,41),(24,47),(25,28),(25,29),(25,36),(25,42),(25,48),(26,57),(26,58),(26,59),(26,60),(27,49),(27,53),(27,67),(28,50),(28,54),(28,67),(29,51),(29,55),(29,67),(30,52),(30,56),(30,67),(31,63),(31,67),(32,61),(32,67),(33,62),(33,67),(34,64),(34,67),(35,65),(35,67),(36,66),(36,67),(37,49),(37,52),(37,63),(38,49),(38,50),(38,61),(39,49),(39,51),(39,62),(40,50),(40,52),(40,64),(41,51),(41,52),(41,65),(42,50),(42,51),(42,66),(43,53),(43,56),(43,63),(44,53),(44,54),(44,61),(45,53),(45,55),(45,62),(46,54),(46,56),(46,64),(47,55),(47,56),(47,65),(48,54),(48,55),(48,66),(49,68),(50,68),(51,68),(52,68),(53,68),(54,68),(55,68),(56,68),(57,61),(57,63),(57,64),(58,62),(58,63),(58,65),(59,61),(59,62),(59,66),(60,64),(60,65),(60,66),(61,68),(62,68),(63,68),(64,68),(65,68),(66,68),(67,68)],69)
 => ? = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 4 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(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),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,10),(1,11),(1,40),(1,41),(2,15),(2,16),(2,17),(2,26),(2,27),(2,41),(3,12),(3,13),(3,14),(3,24),(3,25),(3,41),(4,21),(4,22),(4,23),(4,25),(4,27),(4,40),(5,18),(5,19),(5,20),(5,24),(5,26),(5,40),(6,11),(6,14),(6,17),(6,20),(6,23),(6,42),(7,10),(7,13),(7,16),(7,19),(7,22),(7,42),(8,9),(8,12),(8,15),(8,18),(8,21),(8,42),(9,43),(9,46),(9,49),(10,44),(10,47),(10,49),(11,45),(11,48),(11,49),(12,28),(12,31),(12,43),(12,50),(13,29),(13,32),(13,44),(13,50),(14,30),(14,33),(14,45),(14,50),(15,34),(15,37),(15,43),(15,51),(16,35),(16,38),(16,44),(16,51),(17,36),(17,39),(17,45),(17,51),(18,28),(18,34),(18,46),(18,52),(19,29),(19,35),(19,47),(19,52),(20,30),(20,36),(20,48),(20,52),(21,31),(21,37),(21,46),(21,53),(22,32),(22,38),(22,47),(22,53),(23,33),(23,39),(23,48),(23,53),(24,28),(24,29),(24,30),(24,58),(25,31),(25,32),(25,33),(25,58),(26,34),(26,35),(26,36),(26,58),(27,37),(27,38),(27,39),(27,58),(28,54),(28,61),(29,54),(29,62),(30,54),(30,63),(31,55),(31,61),(32,55),(32,62),(33,55),(33,63),(34,56),(34,61),(35,56),(35,62),(36,56),(36,63),(37,57),(37,61),(38,57),(38,62),(39,57),(39,63),(40,46),(40,47),(40,48),(40,58),(41,43),(41,44),(41,45),(41,58),(42,49),(42,50),(42,51),(42,52),(42,53),(43,59),(43,61),(44,59),(44,62),(45,59),(45,63),(46,60),(46,61),(47,60),(47,62),(48,60),(48,63),(49,59),(49,60),(50,54),(50,55),(50,59),(51,56),(51,57),(51,59),(52,54),(52,56),(52,60),(53,55),(53,57),(53,60),(54,64),(55,64),(56,64),(57,64),(58,61),(58,62),(58,63),(59,64),(60,64),(61,64),(62,64),(63,64)],65)
 => ? = 4 - 2
([(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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,17),(1,23),(1,30),(1,42),(1,44),(2,10),(2,16),(2,22),(2,30),(2,41),(2,43),(3,12),(3,18),(3,24),(3,29),(3,41),(3,44),(4,13),(4,19),(4,25),(4,29),(4,42),(4,43),(5,15),(5,21),(5,27),(5,28),(5,43),(5,44),(6,14),(6,20),(6,26),(6,28),(6,41),(6,42),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(7,31),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,31),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,31),(10,32),(10,45),(10,47),(10,57),(11,32),(11,46),(11,48),(11,58),(12,33),(12,45),(12,48),(12,59),(13,33),(13,46),(13,47),(13,60),(14,34),(14,45),(14,46),(14,61),(15,34),(15,47),(15,48),(15,62),(16,35),(16,49),(16,51),(16,57),(17,35),(17,50),(17,52),(17,58),(18,36),(18,49),(18,52),(18,59),(19,36),(19,50),(19,51),(19,60),(20,37),(20,49),(20,50),(20,61),(21,37),(21,51),(21,52),(21,62),(22,38),(22,53),(22,55),(22,57),(23,38),(23,54),(23,56),(23,58),(24,39),(24,53),(24,56),(24,59),(25,39),(25,54),(25,55),(25,60),(26,40),(26,53),(26,54),(26,61),(27,40),(27,55),(27,56),(27,62),(28,34),(28,37),(28,40),(28,70),(29,33),(29,36),(29,39),(29,70),(30,32),(30,35),(30,38),(30,70),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(32,63),(32,71),(33,64),(33,71),(34,65),(34,71),(35,63),(35,72),(36,64),(36,72),(37,65),(37,72),(38,63),(38,73),(39,64),(39,73),(40,65),(40,73),(41,45),(41,49),(41,53),(41,70),(42,46),(42,50),(42,54),(42,70),(43,47),(43,51),(43,55),(43,70),(44,48),(44,52),(44,56),(44,70),(45,66),(45,71),(46,67),(46,71),(47,68),(47,71),(48,69),(48,71),(49,66),(49,72),(50,67),(50,72),(51,68),(51,72),(52,69),(52,72),(53,66),(53,73),(54,67),(54,73),(55,68),(55,73),(56,69),(56,73),(57,63),(57,66),(57,68),(58,63),(58,67),(58,69),(59,64),(59,66),(59,69),(60,64),(60,67),(60,68),(61,65),(61,66),(61,67),(62,65),(62,68),(62,69),(63,74),(64,74),(65,74),(66,74),(67,74),(68,74),(69,74),(70,71),(70,72),(70,73),(71,74),(72,74),(73,74)],75)
 => ? = 4 - 2
([],7)
 => ([],1)
 => ([],1)
 => 1 = 3 - 2
([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(3,6),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 3 - 2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 3 - 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 1 = 3 - 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 3 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 3 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 3 - 2
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 3 - 2
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
 => ? = 4 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
 => ? = 3 - 2
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,11),(1,12),(1,16),(1,20),(2,8),(2,9),(2,12),(2,15),(2,19),(3,7),(3,9),(3,11),(3,14),(3,18),(4,7),(4,8),(4,10),(4,13),(4,17),(5,17),(5,18),(5,19),(5,20),(5,21),(6,13),(6,14),(6,15),(6,16),(6,21),(7,24),(7,30),(7,44),(8,22),(8,28),(8,44),(9,23),(9,29),(9,44),(10,25),(10,31),(10,44),(11,26),(11,32),(11,44),(12,27),(12,33),(12,44),(13,22),(13,24),(13,25),(13,34),(14,23),(14,24),(14,26),(14,35),(15,22),(15,23),(15,27),(15,36),(16,25),(16,26),(16,27),(16,37),(17,28),(17,30),(17,31),(17,34),(18,29),(18,30),(18,32),(18,35),(19,28),(19,29),(19,33),(19,36),(20,31),(20,32),(20,33),(20,37),(21,34),(21,35),(21,36),(21,37),(22,38),(22,45),(23,39),(23,45),(24,40),(24,45),(25,41),(25,45),(26,42),(26,45),(27,43),(27,45),(28,38),(28,46),(29,39),(29,46),(30,40),(30,46),(31,41),(31,46),(32,42),(32,46),(33,43),(33,46),(34,38),(34,40),(34,41),(35,39),(35,40),(35,42),(36,38),(36,39),(36,43),(37,41),(37,42),(37,43),(38,47),(39,47),(40,47),(41,47),(42,47),(43,47),(44,45),(44,46),(45,47),(46,47)],48)
 => ? = 3 - 2

Description

The size of the image of the pop stack sorting operator. The pop stack sorting operator is defined by

$Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$ . This statistic returns the size of

$Pop_L^\downarrow(L)\}$ .

The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001846The number of elements which do not have a complement in the lattice. St001845The number of join irreducibles minus the rank of a lattice. St001651The Frankl number of a lattice. St000068The number of minimal elements in a poset. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001330The hat guessing number of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001703The villainy of a graph. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001490The number of connected components of a skew partition.