Identifier
-
Mp00223:
Permutations
—runsort⟶
Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000422: Graphs ⟶ ℤ
Values
[1] => [1] => ([],1) => ([],1) => 0
[1,2] => [1,2] => ([(0,1)],2) => ([(0,1)],2) => 2
[2,1] => [1,2] => ([(0,1)],2) => ([(0,1)],2) => 2
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 4
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 4
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 4
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 4
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,2,6,4,3,5] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,5,2,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,5,3,4,6,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,5,4,6,2,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,5,4,6,3,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,1,4,5,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,1,5,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,1,5,4,6,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,3,1,5,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,3,4,6,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,3,6,1,4,5] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,6,1,3,4,5] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[2,6,3,5,4,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[2,6,4,3,5,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,1,5,4,6,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[3,2,1,5,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[3,4,6,1,5,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[3,4,6,2,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[3,5,1,2,6,4] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,5,2,6,4,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,5,4,1,2,6] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,5,4,2,6,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[3,6,1,4,5,2] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[3,6,2,1,4,5] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[4,1,2,6,3,5] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[4,2,6,3,5,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[4,3,5,1,2,6] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[4,3,5,2,6,1] => [1,2,6,3,5,4] => ([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 6
[4,6,1,5,2,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[4,6,1,5,3,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[4,6,2,1,5,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[4,6,2,3,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[4,6,3,1,5,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[4,6,3,2,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,3,4,5,2] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,4,5,2,3] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,4,5,3,2] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,5,2,3,4] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,5,3,4,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,5,4,2,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,1,5,4,3,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,1,3,4,5] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,1,4,5,3] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,1,5,3,4] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,1,5,4,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,3,1,4,5] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,3,1,5,4] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,2,3,4,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,3,1,4,5,2] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,3,1,5,4,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,3,2,1,4,5] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,3,2,1,5,4] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,3,4,1,5,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,3,4,2,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,4,1,5,2,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,4,1,5,3,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,4,2,1,5,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,4,2,3,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,4,3,1,5,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[6,4,3,2,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 8
[1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,2,3,6,7,5,4] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,7,2,5,6,3,4] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,7,2,5,6,4,3] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,7,3,2,5,6,4] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,7,3,4,2,5,6] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,7,4,2,5,6,3] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[1,7,4,3,2,5,6] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,3,6,7,4,5,1] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,3,6,7,5,4,1] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,1,7,3,4] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,1,7,4,3] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,3,1,7,4] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,3,4,1,7] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,4,1,7,3] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[2,5,6,4,3,1,7] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,1,7,2,5,6,4] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,1,7,4,2,5,6] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,2,5,6,1,7,4] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,2,5,6,4,1,7] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,4,1,7,2,5,6] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,4,2,5,6,1,7] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,6,7,4,5,1,2] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,6,7,4,5,2,1] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,6,7,5,4,1,2] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,2,3,6,7,5] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,7,2,5,6,3] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,1,7,3,2,5,6] => [1,7,2,5,6,3,4] => ([(0,3),(0,6),(4,2),(5,1),(6,4),(6,5)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
[4,2,3,6,7,5,1] => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 8
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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.
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.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
to graph
Description
Returns the Hasse diagram of the poset as an undirected graph.
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