Processing math: 100%

Identifier
Values
[1] => [1,0,1,0] => [1,1] => ([(0,1)],2) => 2
[1,1] => [1,0,1,1,0,0] => [1,2] => ([(1,2)],3) => 2
[2,1] => [1,0,1,0,1,0] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 4
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,3] => ([(2,3)],4) => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3] => ([(2,3)],4) => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => ([(3,4)],5) => 2
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 4
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,4] => ([(3,4)],5) => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,5] => ([(4,5)],6) => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4] => ([(3,4)],5) => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,4] => ([(3,4)],5) => 2
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 2
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,4] => ([(3,4)],5) => 2
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 4
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 2
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 2
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 4
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 2
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 2
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 2
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => ([(4,5)],6) => 2
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 2
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 4
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => ([(4,5)],6) => 2
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 2
[4,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => ([(4,5)],6) => 2
[3,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 4
[3,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[5,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => ([(4,5)],6) => 2
[4,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 4
[4,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 4
[3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[5,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[4,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 4
[4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[3,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[5,4,4] => [1,1,1,1,0,0,0,0,1,1,0,1,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[5,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,4,1] => [1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[4,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
[4,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[3,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,6] => ([(5,6)],7) => 2
[3,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[5,4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,1,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[4,4,4,2] => [1,1,1,0,0,1,0,0,1,1,1,0,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[4,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8
[4,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
>>> Load all 132 entries. <<<
[3,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[5,4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,1,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[5,4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10
[5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[4,4,4,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[4,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[4,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [1,6] => ([(5,6)],7) => 2
[4,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[4,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[3,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[5,4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,1,0,0] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[5,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
[5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,6] => ([(5,6)],7) => 2
[5,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[5,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[4,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[4,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[4,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[5,4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 4
[5,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[4,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
[6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 12
[5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 10
[5,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 8
[4,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 8
[5,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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.
Map
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.