- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤ

Values

[1,0] => [1] => ([],1) => 1

[1,0,1,0] => [1,1] => ([(0,1)],2) => 2

[1,1,0,0] => [2] => ([],2) => 1

[1,0,1,0,1,0] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3

[1,0,1,1,0,0] => [1,2] => ([(1,2)],3) => 2

[1,1,0,0,1,0] => [2,1] => ([(0,2),(1,2)],3) => 2

[1,1,0,1,0,0] => [3] => ([],3) => 1

[1,1,1,0,0,0] => [3] => ([],3) => 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) => 4

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

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

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

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

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

[1,1,0,1,0,1,0,0] => [4] => ([],4) => 1

[1,1,0,1,1,0,0,0] => [4] => ([],4) => 1

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

[1,1,1,0,0,1,0,0] => [4] => ([],4) => 1

[1,1,1,0,1,0,0,0] => [4] => ([],4) => 1

[1,1,1,1,0,0,0,0] => [4] => ([],4) => 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) => 5

[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) => 4

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,1,0,0] => [5] => ([],5) => 1

[1,1,0,1,0,1,1,0,0,0] => [5] => ([],5) => 1

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

[1,1,0,1,1,0,0,1,0,0] => [5] => ([],5) => 1

[1,1,0,1,1,0,1,0,0,0] => [5] => ([],5) => 1

[1,1,0,1,1,1,0,0,0,0] => [5] => ([],5) => 1

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

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

[1,1,1,0,0,1,0,1,0,0] => [5] => ([],5) => 1

[1,1,1,0,0,1,1,0,0,0] => [5] => ([],5) => 1

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

[1,1,1,0,1,0,0,1,0,0] => [5] => ([],5) => 1

[1,1,1,0,1,0,1,0,0,0] => [5] => ([],5) => 1

[1,1,1,0,1,1,0,0,0,0] => [5] => ([],5) => 1

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

[1,1,1,1,0,0,0,1,0,0] => [5] => ([],5) => 1

[1,1,1,1,0,0,1,0,0,0] => [5] => ([],5) => 1

[1,1,1,1,0,1,0,0,0,0] => [5] => ([],5) => 1

[1,1,1,1,1,0,0,0,0,0] => [5] => ([],5) => 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) => 6

[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) => 5

[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) => 4

[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) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,1,0,1,0,0] => [6] => ([],6) => 1

[1,1,0,1,0,1,0,1,1,0,0,0] => [6] => ([],6) => 1

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

[1,1,0,1,0,1,1,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,0,1,0,1,1,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,0,1,0,1,1,1,0,0,0,0] => [6] => ([],6) => 1

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

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

[1,1,0,1,1,0,0,1,0,1,0,0] => [6] => ([],6) => 1

[1,1,0,1,1,0,0,1,1,0,0,0] => [6] => ([],6) => 1

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

[1,1,0,1,1,0,1,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,0,1,1,0,1,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,0,1,1,0,1,1,0,0,0,0] => [6] => ([],6) => 1

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

[1,1,0,1,1,1,0,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,0,1,1,1,0,0,1,0,0,0] => [6] => ([],6) => 1

>>> Load all 322 entries. <<<

[1,1,0,1,1,1,0,1,0,0,0,0] => [6] => ([],6) => 1

[1,1,0,1,1,1,1,0,0,0,0,0] => [6] => ([],6) => 1

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

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

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

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

[1,1,1,0,0,1,0,1,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,0,0,1,0,1,1,0,0,0] => [6] => ([],6) => 1

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

[1,1,1,0,0,1,1,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,0,0,1,1,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,1,0,0,1,1,1,0,0,0,0] => [6] => ([],6) => 1

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

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

[1,1,1,0,1,0,0,1,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,0,1,0,0,1,1,0,0,0] => [6] => ([],6) => 1

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

[1,1,1,0,1,0,1,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,0,1,0,1,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,1,0,1,0,1,1,0,0,0,0] => [6] => ([],6) => 1

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

[1,1,1,0,1,1,0,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,0,1,1,0,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,1,0,1,1,0,1,0,0,0,0] => [6] => ([],6) => 1

[1,1,1,0,1,1,1,0,0,0,0,0] => [6] => ([],6) => 1

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

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

[1,1,1,1,0,0,0,1,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,1,0,0,0,1,1,0,0,0] => [6] => ([],6) => 1

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

[1,1,1,1,0,0,1,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,1,0,0,1,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,1,1,0,0,1,1,0,0,0,0] => [6] => ([],6) => 1

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

[1,1,1,1,0,1,0,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,1,0,1,0,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,1,1,0,1,0,1,0,0,0,0] => [6] => ([],6) => 1

[1,1,1,1,0,1,1,0,0,0,0,0] => [6] => ([],6) => 1

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

[1,1,1,1,1,0,0,0,0,1,0,0] => [6] => ([],6) => 1

[1,1,1,1,1,0,0,0,1,0,0,0] => [6] => ([],6) => 1

[1,1,1,1,1,0,0,1,0,0,0,0] => [6] => ([],6) => 1

[1,1,1,1,1,0,1,0,0,0,0,0] => [6] => ([],6) => 1

[1,1,1,1,1,1,0,0,0,0,0,0] => [6] => ([],6) => 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) => 7

[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) => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => 8

[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 2

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + 2\ q^{2} + q^{3}$

Description

The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

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.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.