- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000081: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [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) => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 354 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [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) => 10

[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [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) => 10

[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [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) => 10

[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [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) => 10

[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [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) => 10

[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [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) => 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [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) => 10

[[1,5,7],[2],[3],[4],[6]] => [6,4,3,2,1,5,7] => [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) => 10

[[1,4,7],[2],[3],[5],[6]] => [6,5,3,2,1,4,7] => [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) => 10

[[1,3,7],[2],[4],[5],[6]] => [6,5,4,2,1,3,7] => [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) => 10

[[1,2,7],[3],[4],[5],[6]] => [6,5,4,3,1,2,7] => [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) => 10

[[1,5,6],[2],[3],[4],[7]] => [7,4,3,2,1,5,6] => [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) => 10

[[1,4,6],[2],[3],[5],[7]] => [7,5,3,2,1,4,6] => [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) => 10

[[1,3,6],[2],[4],[5],[7]] => [7,5,4,2,1,3,6] => [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) => 10

[[1,2,6],[3],[4],[5],[7]] => [7,5,4,3,1,2,6] => [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) => 10

[[1,4,5],[2],[3],[6],[7]] => [7,6,3,2,1,4,5] => [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) => 10

[[1,3,5],[2],[4],[6],[7]] => [7,6,4,2,1,3,5] => [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) => 10

[[1,2,5],[3],[4],[6],[7]] => [7,6,4,3,1,2,5] => [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) => 10

[[1,3,4],[2],[5],[6],[7]] => [7,6,5,2,1,3,4] => [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) => 10

[[1,2,4],[3],[5],[6],[7]] => [7,6,5,3,1,2,4] => [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) => 10

[[1,2,3],[4],[5],[6],[7]] => [7,6,5,4,1,2,3] => [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) => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,6],[2,7],[3],[4],[5]] => [5,4,3,2,7,1,6] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,4],[2,7],[3],[5],[6]] => [6,5,3,2,7,1,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,3],[2,7],[4],[5],[6]] => [6,5,4,2,7,1,3] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,2],[3,7],[4],[5],[6]] => [6,5,4,3,7,1,2] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,5],[2,6],[3],[4],[7]] => [7,4,3,2,6,1,5] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,4],[2,6],[3],[5],[7]] => [7,5,3,2,6,1,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,3],[2,6],[4],[5],[7]] => [7,5,4,2,6,1,3] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,2],[3,6],[4],[5],[7]] => [7,5,4,3,6,1,2] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,4],[2,5],[3],[6],[7]] => [7,6,3,2,5,1,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,3],[2,5],[4],[6],[7]] => [7,6,4,2,5,1,3] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,2],[3,5],[4],[6],[7]] => [7,6,4,3,5,1,2] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,3],[2,4],[5],[6],[7]] => [7,6,5,2,4,1,3] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,2],[3,4],[5],[6],[7]] => [7,6,5,3,4,1,2] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 11

[[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [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) => 15

[[1,6],[2],[3],[4],[5],[7]] => [7,5,4,3,2,1,6] => [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) => 15

[[1,5],[2],[3],[4],[6],[7]] => [7,6,4,3,2,1,5] => [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) => 15

[[1,4],[2],[3],[5],[6],[7]] => [7,6,5,3,2,1,4] => [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) => 15

[[1,3],[2],[4],[5],[6],[7]] => [7,6,5,4,2,1,3] => [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) => 15

[[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [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) => 15

[[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [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) => 21

[[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [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) => 28

[[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9) => 36

[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10) => 45

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Description

The number of edges of a graph.

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descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.

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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.

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reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.