- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001563: Integer partitions ⟶ ℤ

Values

[1,2] => [.,[.,.]] => [1,0,1,0] => [1] => 1

[1,2,3] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => [2,1] => 4

[1,3,2] => [.,[[.,.],.]] => [1,0,1,1,0,0] => [1,1] => 4

[2,1,3] => [[.,.],[.,.]] => [1,1,0,0,1,0] => [2] => 1

[2,3,1] => [[.,[.,.]],.] => [1,1,0,1,0,0] => [1] => 1

[3,1,2] => [[.,.],[.,.]] => [1,1,0,0,1,0] => [2] => 1

[1,2,3,4] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [3,2,1] => 27

[1,2,4,3] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [2,2,1] => 27

[1,3,2,4] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [3,1,1] => 27

[1,3,4,2] => [.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [2,1,1] => 27

[1,4,2,3] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [3,1,1] => 27

[1,4,3,2] => [.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [1,1,1] => 27

[2,1,3,4] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [3,2] => 4

[2,1,4,3] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [2,2] => 4

[2,3,1,4] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [3,1] => 4

[2,3,4,1] => [[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [2,1] => 4

[2,4,1,3] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [3,1] => 4

[2,4,3,1] => [[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [1,1] => 4

[3,1,2,4] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [3,2] => 4

[3,1,4,2] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [2,2] => 4

[3,2,1,4] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [3] => 1

[3,2,4,1] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [2] => 1

[3,4,1,2] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [3,1] => 4

[3,4,2,1] => [[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => [1] => 1

[4,1,2,3] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [3,2] => 4

[4,1,3,2] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [2,2] => 4

[4,2,1,3] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [3] => 1

[4,2,3,1] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [2] => 1

[4,3,1,2] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [3] => 1

[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 256

[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => 256

[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => 256

[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 256

[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 256

[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => 256

[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 256

[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => 256

[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 256

[1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 256

[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 27

[2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 27

[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 27

[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 27

[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => [3,2,1] => 27

[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 27

[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => [2,2,1] => 27

[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 27

[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 27

[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 27

[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 27

[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.] => [1,1,0,1,1,0,1,0,0,0] => [2,1,1] => 27

[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 27

[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 27

[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => [3,1,1] => 27

[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 27

[2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1] => 27

[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 27

[3,1,5,4,2] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 27

[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 4

[3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 4

[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => 4

[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 4

[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 27

[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 4

[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => 4

[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [4,2,1] => 27

[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => [1,1,1,0,1,0,1,0,0,0] => [2,1] => 4

[3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 27

[3,5,2,1,4] => [[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 4

[3,5,2,4,1] => [[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => 4

[3,5,4,1,2] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [4,1,1] => 27

[3,5,4,2,1] => [[[.,[[.,.],.]],.],.] => [1,1,1,0,1,1,0,0,0,0] => [1,1] => 4

[4,1,3,5,2] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 27

[4,1,5,3,2] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 27

[4,2,1,3,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 4

[4,2,1,5,3] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 4

[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[4,2,3,5,1] => [[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => 4

[4,2,5,1,3] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[4,2,5,3,1] => [[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 4

[4,3,1,2,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 4

[4,3,1,5,2] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 4

[4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1

[4,3,2,5,1] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [3] => 1

[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[4,3,5,2,1] => [[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => [2] => 1

[4,5,1,3,2] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [3,3,1] => 27

[4,5,2,1,3] => [[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 4

[4,5,2,3,1] => [[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => 4

[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 4

[4,5,3,2,1] => [[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => [1] => 1

[5,1,3,4,2] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [3,2,2] => 27

[5,1,4,3,2] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 27

[5,2,1,3,4] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 4

[5,2,1,4,3] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 4

[5,2,3,1,4] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => 4

[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[5,2,4,3,1] => [[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => [2,2] => 4

>>> Load all 514 entries. <<<

[5,3,1,2,4] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 4

[5,3,1,4,2] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 4

[5,3,2,1,4] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1

[5,3,2,4,1] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [3] => 1

[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [4,2] => 4

[5,3,4,2,1] => [[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => [2] => 1

[5,4,1,2,3] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 4

[5,4,1,3,2] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 4

[5,4,2,1,3] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1

[5,4,2,3,1] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [3] => 1

[5,4,3,1,2] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1

[1,4,6,5,3,2] => [.,[[[.,[[.,.],.]],.],.]] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => 3125

[1,5,4,6,3,2] => [.,[[[[.,.],[.,.]],.],.]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => 3125

[1,5,6,4,3,2] => [.,[[[[.,[.,.]],.],.],.]] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => 3125

[1,6,4,5,3,2] => [.,[[[[.,.],[.,.]],.],.]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => 3125

[1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 3125

[2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.] => [1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => 256

[2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => 256

[2,4,6,5,3,1] => [[.,[[.,[[.,.],.]],.]],.] => [1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => 256

[2,5,4,3,6,1] => [[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => 256

[2,5,4,6,3,1] => [[.,[[[.,.],[.,.]],.]],.] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => 256

[2,5,6,4,3,1] => [[.,[[[.,[.,.]],.],.]],.] => [1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => 256

[2,6,4,3,5,1] => [[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => 256

[2,6,4,5,3,1] => [[.,[[[.,.],[.,.]],.]],.] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => 256

[2,6,5,3,4,1] => [[.,[[[.,.],.],[.,.]]],.] => [1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => 256

[2,6,5,4,3,1] => [[.,[[[[.,.],.],.],.]],.] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => 256

[3,2,5,6,4,1] => [[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => 27

[3,2,6,5,4,1] => [[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 27

[3,4,2,6,5,1] => [[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 27

[3,4,5,2,6,1] => [[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 27

[3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => 27

[3,4,6,2,5,1] => [[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 27

[3,4,6,5,2,1] => [[[.,[.,[[.,.],.]]],.],.] => [1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => 27

[3,5,2,6,4,1] => [[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 27

[3,5,4,2,1,6] => [[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 27

[3,5,4,2,6,1] => [[[.,[[.,.],.]],[.,.]],.] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 27

[3,5,4,6,2,1] => [[[.,[[.,.],[.,.]]],.],.] => [1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => 27

[3,5,6,2,4,1] => [[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 27

[3,5,6,4,2,1] => [[[.,[[.,[.,.]],.]],.],.] => [1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => 27

[3,6,2,5,4,1] => [[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 27

[3,6,4,2,1,5] => [[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 27

[3,6,4,2,5,1] => [[[.,[[.,.],.]],[.,.]],.] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 27

[3,6,4,5,2,1] => [[[.,[[.,.],[.,.]]],.],.] => [1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => 27

[3,6,5,2,1,4] => [[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 27

[3,6,5,2,4,1] => [[[.,[[.,.],.]],[.,.]],.] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 27

[3,6,5,4,2,1] => [[[.,[[[.,.],.],.]],.],.] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => 27

[4,2,5,6,3,1] => [[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => 27

[4,2,6,5,3,1] => [[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 27

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

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

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

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

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

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

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

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

[4,5,2,6,3,1] => [[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 27

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

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

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

[4,5,6,2,3,1] => [[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => 27

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

[4,6,2,5,3,1] => [[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 27

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

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

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

[4,6,5,2,1,3] => [[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 27

[4,6,5,2,3,1] => [[[.,[[.,.],.]],[.,.]],.] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => 27

[4,6,5,3,1,2] => [[[.,[[.,.],.]],.],[.,.]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => 27

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

[5,2,4,6,3,1] => [[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => 27

[5,2,6,4,3,1] => [[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,6,2,4,3,1] => [[[.,[.,.]],[[.,.],.]],.] => [1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => 27

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

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

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

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

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

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

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

[6,2,4,5,3,1] => [[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => 27

[6,2,5,4,3,1] => [[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,6,7,5,4,3,2] => [.,[[[[[.,[.,.]],.],.],.],.]] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => 46656

[1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => 46656

[2,5,7,6,4,3,1] => [[.,[[[.,[[.,.],.]],.],.]],.] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => 3125

[2,6,5,7,4,3,1] => [[.,[[[[.,.],[.,.]],.],.]],.] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => 3125

[2,6,7,5,4,3,1] => [[.,[[[[.,[.,.]],.],.],.]],.] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => 3125

[2,7,5,6,4,3,1] => [[.,[[[[.,.],[.,.]],.],.]],.] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => 3125

[2,7,6,5,4,3,1] => [[.,[[[[[.,.],.],.],.],.]],.] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => 3125

[3,4,7,6,5,2,1] => [[[.,[.,[[[.,.],.],.]]],.],.] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => 256

[3,5,6,7,4,2,1] => [[[.,[[.,[.,[.,.]]],.]],.],.] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => 256

[3,5,7,6,4,2,1] => [[[.,[[.,[[.,.],.]],.]],.],.] => [1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => 256

[3,6,5,4,7,2,1] => [[[.,[[[.,.],.],[.,.]]],.],.] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => 256

[3,6,5,7,4,2,1] => [[[.,[[[.,.],[.,.]],.]],.],.] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => 256

[3,6,7,5,4,2,1] => [[[.,[[[.,[.,.]],.],.]],.],.] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => 256

[3,7,5,4,6,2,1] => [[[.,[[[.,.],.],[.,.]]],.],.] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => 256

[3,7,5,6,4,2,1] => [[[.,[[[.,.],[.,.]],.]],.],.] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => 256

[3,7,6,4,5,2,1] => [[[.,[[[.,.],.],[.,.]]],.],.] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => 256

[3,7,6,5,4,2,1] => [[[.,[[[[.,.],.],.],.]],.],.] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => 256

[4,3,6,7,5,2,1] => [[[[.,.],[[.,[.,.]],.]],.],.] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => 27

[4,3,7,6,5,2,1] => [[[[.,.],[[[.,.],.],.]],.],.] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 27

[4,5,3,7,6,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 27

[4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 27

[4,5,6,7,3,2,1] => [[[[.,[.,[.,[.,.]]]],.],.],.] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => 27

[4,5,7,3,6,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 27

[4,5,7,6,3,2,1] => [[[[.,[.,[[.,.],.]]],.],.],.] => [1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => 27

[4,6,3,7,5,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 27

[4,6,5,3,2,7,1] => [[[[.,[[.,.],.]],.],[.,.]],.] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 27

[4,6,5,3,7,2,1] => [[[[.,[[.,.],.]],[.,.]],.],.] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 27

[4,6,5,7,3,2,1] => [[[[.,[[.,.],[.,.]]],.],.],.] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => 27

[4,6,7,3,5,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 27

[4,6,7,5,3,2,1] => [[[[.,[[.,[.,.]],.]],.],.],.] => [1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => 27

[4,7,3,6,5,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 27

[4,7,5,3,2,6,1] => [[[[.,[[.,.],.]],.],[.,.]],.] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 27

[4,7,5,3,6,2,1] => [[[[.,[[.,.],.]],[.,.]],.],.] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 27

[4,7,5,6,3,2,1] => [[[[.,[[.,.],[.,.]]],.],.],.] => [1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => 27

[4,7,6,3,2,5,1] => [[[[.,[[.,.],.]],.],[.,.]],.] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 27

[4,7,6,3,5,2,1] => [[[[.,[[.,.],.]],[.,.]],.],.] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 27

[4,7,6,5,3,2,1] => [[[[.,[[[.,.],.],.]],.],.],.] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => 27

[5,3,6,7,4,2,1] => [[[[.,.],[[.,[.,.]],.]],.],.] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => 27

[5,3,7,6,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 27

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

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

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

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

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

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

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

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

[5,6,3,7,4,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 27

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

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

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

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

[5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => 27

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

[5,7,3,6,4,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 27

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

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

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

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

[5,7,6,3,2,4,1] => [[[[.,[[.,.],.]],.],[.,.]],.] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 27

[5,7,6,3,4,2,1] => [[[[.,[[.,.],.]],[.,.]],.],.] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => 27

[5,7,6,4,2,3,1] => [[[[.,[[.,.],.]],.],[.,.]],.] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [5,1,1] => 27

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

[6,3,5,7,4,2,1] => [[[[.,.],[[.,[.,.]],.]],.],.] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => 27

[6,3,7,5,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6,7,3,5,4,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.] => [1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => 27

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

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

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

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

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

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

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

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

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

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

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

[7,3,5,6,4,2,1] => [[[[.,.],[[.,[.,.]],.]],.],.] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => 27

[7,3,6,5,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => 27

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

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

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

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

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

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

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

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

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

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

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

[5,6,7,4,8,3,2,1] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,8,7,6,4,3,2,1] => [[[[[.,[[[.,.],.],.]],.],.],.],.] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => 27

[5,7,8,6,4,3,2,1] => [[[[[.,[[.,[.,.]],.]],.],.],.],.] => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => 27

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

[5,6,8,7,4,3,2,1] => [[[[[.,[.,[[.,.],.]]],.],.],.],.] => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => 27

[5,7,6,8,4,3,2,1] => [[[[[.,[[.,.],[.,.]]],.],.],.],.] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => 27

[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => 256

[4,7,8,6,5,3,2,1] => [[[[.,[[[.,[.,.]],.],.]],.],.],.] => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => 256

[4,6,8,7,5,3,2,1] => [[[[.,[[.,[[.,.],.]],.]],.],.],.] => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1] => 256

[4,7,6,8,5,3,2,1] => [[[[.,[[[.,.],[.,.]],.]],.],.],.] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => 256

[4,5,8,7,6,3,2,1] => [[[[.,[.,[[[.,.],.],.]]],.],.],.] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [2,2,2,1] => 256

[5,7,6,4,8,3,2,1] => [[[[[.,[[.,.],.]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => 27

[4,7,6,5,8,3,2,1] => [[[[.,[[[.,.],.],[.,.]]],.],.],.] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1] => 256

[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1] => 3125

[3,7,8,6,5,4,2,1] => [[[.,[[[[.,[.,.]],.],.],.]],.],.] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => 3125

[3,6,8,7,5,4,2,1] => [[[.,[[[.,[[.,.],.]],.],.]],.],.] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1] => 3125

[3,7,6,8,5,4,2,1] => [[[.,[[[[.,.],[.,.]],.],.]],.],.] => [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1] => 3125

[5,7,6,4,3,8,2,1] => [[[[[.,[[.,.],.]],.],[.,.]],.],.] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0] => [5,1,1] => 27

[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => 46656

[2,7,8,6,5,4,3,1] => [[.,[[[[[.,[.,.]],.],.],.],.]],.] => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1,1] => 46656

[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 823543

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,8,7,4,3,6,2,1] => [[[[[.,[[.,.],.]],.],[.,.]],.],.] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0] => [5,1,1] => 27

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

[5,8,6,7,4,3,2,1] => [[[[[.,[[.,.],[.,.]]],.],.],.],.] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => 27

[4,8,7,5,6,3,2,1] => [[[[.,[[[.,.],.],[.,.]]],.],.],.] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [4,1,1,1] => 256

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

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

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

[6,4,7,8,5,3,2,1] => [[[[[.,.],[[.,[.,.]],.]],.],.],.] => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [3,2,2] => 27

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

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

[2,9,8,7,6,5,4,3,1] => [[.,[[[[[[[.,.],.],.],.],.],.],.]],.] => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => 823543

[8,4,7,6,5,3,2,1] => [[[[[.,.],[[[.,.],.],.]],.],.],.] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => 27

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

[5,7,4,8,6,3,2,1] => [[[[[.,[.,.]],[[.,.],.]],.],.],.] => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => 27

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

[6,8,5,9,7,4,3,2,1] => [[[[[[.,[.,.]],[[.,.],.]],.],.],.],.] => [1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1] => 27

[6,7,8,9,5,4,3,2,1] => [[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => 27

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

[7,8,9,10,6,5,4,3,2,1] => [[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => 27

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

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

[5,6,8,4,7,3,2,1] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => 27

[7,4,6,8,5,3,2,1] => [[[[[.,.],[[.,[.,.]],.]],.],.],.] => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [3,2,2] => 27

[4,8,6,7,5,3,2,1] => [[[[.,[[[.,.],[.,.]],.]],.],.],.] => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1] => 256

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

[5,8,4,7,6,3,2,1] => [[[[[.,[.,.]],[[.,.],.]],.],.],.] => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => 27

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

[5,7,8,4,6,3,2,1] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => 27

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

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

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

[7,4,8,6,5,3,2,1] => [[[[[.,.],[[[.,.],.],.]],.],.],.] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => 27

[6,7,4,8,5,3,2,1] => [[[[[.,[.,.]],[[.,.],.]],.],.],.] => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0] => [3,3,1] => 27

[8,4,6,7,5,3,2,1] => [[[[[.,.],[[.,[.,.]],.]],.],.],.] => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0] => [3,2,2] => 27

[6,8,7,9,5,4,3,2,1] => [[[[[[.,[[.,.],[.,.]]],.],.],.],.],.] => [1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => 27

[5,8,7,6,9,4,3,2,1] => [[[[[.,[[[.,.],.],[.,.]]],.],.],.],.] => [1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0] => [4,1,1,1] => 256

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

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

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

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

[7,8,5,9,6,4,3,2,1] => [[[[[[.,[.,.]],[[.,.],.]],.],.],.],.] => [1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0] => [3,3,1] => 27

[6,4,8,7,5,3,2,1] => [[[[[.,.],[[[.,.],.],.]],.],.],.] => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0] => [2,2,2] => 27

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

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

[5,8,6,4,7,3,2,1] => [[[[[.,[[.,.],.]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => 27

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

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

[6,8,7,4,5,3,2,1] => [[[[[.,[[.,.],.]],[.,.]],.],.],.] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => 27

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

[3,8,6,7,5,4,2,1] => [[[.,[[[[.,.],[.,.]],.],.]],.],.] => [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,1,1,1] => 3125

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

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

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

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

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

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

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

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

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

Description

The value of the power-sum symmetric function evaluated at 1.
The statistic is

$p_\lambda(x_1,\dotsc,x_k)$ evaluated at

$x_1=x_2=\dotsb=x_k$ ,
where

$\lambda$ has

$k$ parts.

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition

$\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition

$(n-1,\ldots,2,1)$ when cutting out

$D$ considered as a path from

$(0,0)$ to

$(n,n)$ .
In other words,

$\lambda_{i}$ is the number of down-steps before the

$(n+1-i)$ -th up-step of

$D$ .
This map is a bijection between Dyck paths of size

$n$ and partitions inside the staircase partition

$(n-1,\ldots,2,1)$ .

Map

to Dyck path: up step, left tree, down step, right tree

Description

Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:

a leaf is associated to the empty Dyck Word
a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$ .

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.