- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 296 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

$F_{4} = 1 + 4\ q + 4\ q^{2} + q^{3}$

$F_{5} = 1 + 6\ q + 12\ q^{2} + 6\ q^{3} + q^{4}$

$F_{6} = 1 + 9\ q + 28\ q^{2} + 28\ q^{3} + 9\ q^{4} + q^{5}$

Description

The jump number of the poset.
A jump in a linear extension

$e_1, \dots, e_n$ of a poset

$P$ is a pair

$(e_i, e_{i+1})$ so that

$e_{i+1}$ does not cover

$e_i$ in

$P$ . The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

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.

Map

permutation poset

Description

Sends a permutation to its permutation poset.
For a permutation

$\pi$ of length

$n$ , this poset has vertices

$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$
and the cover relation is given by

$(w, x) \leq (y, z)$ if

$w \leq y$ and

$x \leq z$ .
For example, the permutation

$[3,1,5,4,2]$ is mapped to the poset with cover relations

$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$