- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2160 entries. <<<

Map

to 312-avoiding permutation

Description

Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.