- FindStat

Identifier

Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 187 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

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

Description

The number of two-by-two squares inside a skew partition.
This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside 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

$\gamma(D)$ .