- FindStat

Identifier

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[.,.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[[[[.,.],.],.],.],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 187 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[[.,.],.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[[[[[.,.],.],.],.],.]]] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1],[]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[[[[[.,.],.],.],.],.],.]] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1],[]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,[[[[[.,.],.],.],.],.]],.] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [[6,2],[1]] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[[.,[.,.]],.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[6,6],[5]] => 1

[[[[[[[.,.],.],.],.],.],.],.] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7],[]] => 1

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 8\ q + 6\ q^{2}$

Description

The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.

Map

pruning number to logarithmic height

Description

Francon's map from binary trees to Dyck paths.
This bijection sends the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree., to the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path.. The implementation is a literal translation of Knuth's [2].

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .