- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000069: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,8),(1,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,7)],9) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => ([(0,9),(1,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,8)],10) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => ([(0,2),(0,7),(1,7),(1,8),(2,6),(2,8),(6,4),(6,5),(7,3),(7,6),(8,3),(8,4),(8,5)],9) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 435 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[8,7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => ([(0,8),(1,8),(2,8),(4,6),(5,4),(6,3),(7,5),(8,7)],9) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,2,2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => ([(0,2),(0,7),(1,6),(1,7),(2,5),(2,6),(3,8),(4,8),(5,8),(6,3),(6,4),(7,3),(7,4),(7,5)],9) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => ([(0,6),(1,8),(2,8),(4,5),(5,7),(6,4),(7,1),(7,2),(8,3)],9) => 1

[4,3,3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => ([(0,3),(0,7),(1,6),(1,7),(3,4),(3,5),(3,6),(4,8),(5,8),(6,8),(7,4),(7,5),(8,2)],9) => 1

[4,4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => ([(0,6),(2,8),(3,8),(4,7),(5,1),(6,4),(7,2),(7,3),(8,5)],9) => 1

[6,6,6,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => ([(0,7),(2,9),(3,9),(4,1),(5,4),(6,5),(7,8),(8,2),(8,3),(9,6)],10) => 1

[7,6,6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => ([(0,8),(1,3),(1,4),(3,8),(4,8),(5,6),(6,2),(7,5),(8,7)],9) => 1

[7,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9) => 8

[8,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10) => 9

[8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1

[9,8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10) => 1

[8,6,5,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 1

[6,5,5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => ([(0,6),(0,7),(1,3),(1,6),(1,7),(3,8),(4,2),(5,4),(6,8),(7,8),(8,5)],9) => 1

[5,4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => ([(0,5),(0,6),(0,7),(1,3),(1,7),(3,5),(3,6),(4,2),(5,8),(6,8),(7,8),(8,4)],9) => 1

[9,9,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10)],11) => 10

[10,9,8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11) => 1

[7,7,5,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(6,8),(7,8)],9) => 6

[6,6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(8,1),(8,2)],9) => 7

[7,6,5,4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(6,8),(7,8)],9) => 1

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

click to show known generating functions

$F_{1} = q^{2}$

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

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

$F_{4} = 2\ q + 2\ q^{2} + q^{3}$

$F_{5} = 2\ q + 3\ q^{2} + 2\ q^{3}$

$F_{6} = 4\ q + 4\ q^{2} + 2\ q^{3} + q^{4}$

Description

The number of maximal elements of a poset.

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

Map

Hessenberg poset

Description

The Hessenberg poset of a Dyck path.
Let

$D$ be a Dyck path of semilength

$n$ , regarded as a subdiagonal path from

$(0,0)$ to

$(n,n)$ , and let

$\boldsymbol{m}_i$ be the

$x$ -coordinate of the

$i$ -th up step.
Then the Hessenberg poset (or natural unit interval order) corresponding to

$D$ has elements

$\{1,\dots,n\}$ with

$i < j$ if

$j < \boldsymbol{m}_i$ .

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.