Processing math: 100%

Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000929
Mapping
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1] => [[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,1,2] => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[2,1,1] => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,1,3] => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[3,1,1] => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,2,2] => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[2,2,1] => [[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 1
[1,1,1,1] => [[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,1,1,2] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,2,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,2,1,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,1,1,1] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,1,3] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,3,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,3,1,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[3,1,1,1] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,1,4] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,4,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,4,1,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[4,1,1,1] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,2,2] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,2,1,2] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[1,2,2,1] => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,1,1,2] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,1,2,1] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[2,2,1,1] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,2,3] => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 0
[1,1,3,2] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,2,3,1] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[2,1,1,3] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,3,1,1] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[3,1,1,2] => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[3,2,1,1] => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 0
[1,1,2,4] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,1,4,2] => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 0
[1,4,2,1] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[2,1,1,4] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[2,4,1,1] => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 0
[4,1,1,2] => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[4,2,1,1] => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[1,1,3,3] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,3,1,3] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[1,3,3,1] => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[3,1,1,3] => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[3,1,3,1] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[3,3,1,1] => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 0
[1,1,3,4] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,1,4,3] => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1
[1,3,4,1] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
[1,4,3,1] => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1
Description
The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is 0 for partitions λ≠1n and 1 for λ=1n.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
Mapping
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 2
[1,1,2] => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[2,1,1] => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,1,3] => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[3,1,1] => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,2,2] => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[2,2,1] => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[1,1,1,1] => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,1,1,2] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,2,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,2,1,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[2,1,1,1] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,1,3] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,3,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,3,1,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[3,1,1,1] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,1,4] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,4,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,4,1,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[4,1,1,1] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,2,2] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,1,2] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[1,2,2,1] => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 2
[2,1,1,2] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[2,1,2,1] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,2,1,1] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,2,3] => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,3,2] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,2,3,1] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,1,1,3] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[2,3,1,1] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[3,1,1,2] => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 2
[3,2,1,1] => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,2,4] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,1,4,2] => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,4,2,1] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,1,1,4] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[2,4,1,1] => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[4,1,1,2] => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 2
[4,2,1,1] => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 2
[1,1,3,3] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,3,1,3] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[1,3,3,1] => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 2
[3,1,1,3] => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 0 + 2
[3,1,3,1] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[3,3,1,1] => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,1,3,4] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,1,4,3] => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 2
[1,3,4,1] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[1,4,3,1] => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[3,1,1,4] => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,3,4] => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[2,4,1,3] => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[3,1,4,2] => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[4,3,2,1] => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,1,1,4,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,4,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,4,1,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,1,2,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,1,2,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,2,1,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,1,3,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,1,3,5] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,1,5,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,1,5,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,1,3,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,1,1,5] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,5,1,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[5,1,1,1,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[5,3,1,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,4,4,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,4,4,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[4,1,1,1,4] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[2,1,2,1,2] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,2,2,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,3,2,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,2,2,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,2,3,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,2,1,3] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,2,3,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,3,2,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,4,1,2,3] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,2,1,4,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,3,1,5] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,3,5,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,3,2,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[5,1,3,2,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,2,4,5,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,5,4,2,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,3,1,3] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,3,3,5] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,1,5,3,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,5,3,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,1,5,1,3] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,3,1,1,5] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[3,3,5,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[5,1,1,3,3] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[5,3,3,1,1] => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,3,4,5,1] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[1,4,1,3,5] => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
Mapping
Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[1,1,1] => [[],[],[]]
 => [[[.,.],.],.]
 => ([(0,2),(1,2)],3)
 => ? = 0 + 6
[1,1,2] => [[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 6
[2,1,1] => [[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 6
[1,1,3] => [[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 6
[3,1,1] => [[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 6
[1,2,2] => [[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 6
[2,2,1] => [[],[[]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 6
[1,1,1,1] => [[],[],[],[]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,1,2] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,2,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,2,1,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[2,1,1,1] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,1,3] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,3,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,3,1,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[3,1,1,1] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,1,4] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,4,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,4,1,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[4,1,1,1] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,2,2] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,2,1,2] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[1,2,2,1] => [[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[2,1,1,2] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[2,1,2,1] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[2,2,1,1] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,2,3] => [[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,3,2] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,2,3,1] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[2,1,1,3] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[2,3,1,1] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[3,1,1,2] => [[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[3,2,1,1] => [[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,2,4] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,1,4,2] => [[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,4,2,1] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[2,1,1,4] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[2,4,1,1] => [[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[4,1,1,2] => [[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[4,2,1,1] => [[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,3,3] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,3,1,3] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[1,3,3,1] => [[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[3,1,1,3] => [[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[3,1,3,1] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[3,3,1,1] => [[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 6
[1,1,3,4] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,1,4,3] => [[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 6
[1,3,4,1] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[1,4,3,1] => [[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 6
[1,2,1,2,1,3] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,1,2,1,4] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,1,2,1,5] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,1,2,5,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,1,2,1,6] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,3,1,3,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,1,3,1,2] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,1,3,2,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,3,1,6,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,6,1,2,3,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,1,4,1,2] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,1,4,5,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,5,1,2,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,2,1,4,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,1,5,1,2] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,1,5,2,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,1,3,1,4] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,1,3,1,5] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,1,3,5,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,5,1,3,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,6,1,3,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,1,4,1,3] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,5,1,4,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,1,3,5,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,1,4,6,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,6,1,4,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,1,3,6,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,3,5,1,5,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,1,5,1,3] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,1,5,3,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,6,3,1,5,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,1,4,5,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,5,1,4,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,4,1,4,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,1,4,1,6] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,4,5,1,5,1] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,5,1,6,4,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,6,4,1,5,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[2,1,2,1,3,2] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[2,1,3,2,1,2] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[2,3,1,2,1,2] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[2,5,1,2,1,2] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,1,3,2,3] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,3,2,1,3] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[2,3,1,3,1,2] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[3,2,1,2,1,3] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,3,1,2,4] => [[[],[[]],[[]]]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,3,1,4,2] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,3,2,1,4] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
[1,2,4,2,3,1] => [[[],[[],[[]]]]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6 = 0 + 6
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 Kn equals 2n−2. For this reason, we do not define the energy of the empty graph.