Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000457
(load all 2 compositions to match this statistic)
Mapping
St000457: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 4
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 2
[2,3,4,1] => 0
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 4
[3,2,4,1] => 2
[3,4,1,2] => 0
[3,4,2,1] => 2
[4,1,2,3] => 0
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 3
[1,2,4,5,3] => 4
[1,2,5,3,4] => 4
[1,2,5,4,3] => 7
[1,3,2,4,5] => 3
[1,3,2,5,4] => 6
[1,3,4,2,5] => 4
[1,3,4,5,2] => 3
[1,3,5,2,4] => 5
[1,3,5,4,2] => 6
[1,4,2,3,5] => 4
[1,4,2,5,3] => 5
[1,4,3,2,5] => 7
[1,4,3,5,2] => 6
[1,4,5,2,3] => 4
Description
The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[1] => [.,.]
 => [[]]
 => ([(0,1)],2)
 => 2 = 0 + 2
[1,2] => [.,[.,.]]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0} + 2
[2,1] => [[.,.],.]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,0} + 2
[1,2,3] => [.,[.,[.,.]]]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1} + 2
[1,3,2] => [.,[[.,.],.]]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1} + 2
[2,1,3] => [[.,.],[.,.]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1} + 2
[2,3,1] => [[.,.],[.,.]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1} + 2
[3,1,2] => [[.,[.,.]],.]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1} + 2
[3,2,1] => [[[.,.],.],.]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,1} + 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 2 + 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4} + 2
[4,3,2,1] => [[[[.,.],.],.],.]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [[[[[[]]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,10,10,10,10,10} + 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[3,5,2,1,4] => [[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 4 + 2
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[[[[]],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[[[[]],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[[[[]],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,1,5,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,1,5,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,1,5,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,1,3,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,1,6,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,3,1,4,6] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,3,1,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,3,4,1,6] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,3,4,6,1] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,3,6,1,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,3,6,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,6,1,3,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,6,3,1,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[2,5,6,3,4,1] => [[.,.],[[.,[.,.]],[.,.]]]
 => [[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,1,2,4,6] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,1,2,6,4] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,1,4,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,1,4,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,1,6,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,1,6,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,4,1,2,6] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,4,1,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,4,6,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,6,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,6,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,3,6,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,6,3,1,2,4] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,6,3,1,4,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
[5,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 6 + 2
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.