- FindStat

Identifier

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001004: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see St000371The number of mid points of decreasing subsequences of length 3 in a permutation. and St000372The number of mid points of increasing subsequences of length 3 in a permutation..

Map

to 132-avoiding permutation

Description

Return a 132-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 maximal element of the Sylvester class.

Map

to binary tree: left brother = left child

Description

Return a binary tree of size $n-1$ (where $n$ is the size of $t$, and where $t$ is an ordered tree) by the following recursive rule:
- if $x$ is the left brother of $y$ in $t$, then $x$ becomes the left child of $y$;
- if $x$ is the last child of $y$ in $t$, then $x$ becomes the right child of $y$,
and removing the root of $t$.