Identifier
-
Mp00049:
Ordered trees
—to binary tree: left brother = left child⟶
Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000991: Permutations ⟶ ℤ
Values
[[]] => [.,.] => [1] => 1
[[],[]] => [[.,.],.] => [1,2] => 2
[[[]]] => [.,[.,.]] => [2,1] => 1
[[],[],[]] => [[[.,.],.],.] => [1,2,3] => 3
[[],[[]]] => [[.,.],[.,.]] => [3,1,2] => 2
[[[]],[]] => [[.,[.,.]],.] => [2,1,3] => 2
[[[],[]]] => [.,[[.,.],.]] => [2,3,1] => 1
[[[[]]]] => [.,[.,[.,.]]] => [3,2,1] => 1
[[],[],[],[]] => [[[[.,.],.],.],.] => [1,2,3,4] => 4
[[],[],[[]]] => [[[.,.],.],[.,.]] => [4,1,2,3] => 3
[[],[[]],[]] => [[[.,.],[.,.]],.] => [3,1,2,4] => 3
[[],[[],[]]] => [[.,.],[[.,.],.]] => [3,4,1,2] => 2
[[],[[[]]]] => [[.,.],[.,[.,.]]] => [4,3,1,2] => 2
[[[]],[],[]] => [[[.,[.,.]],.],.] => [2,1,3,4] => 3
[[[]],[[]]] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 2
[[[],[]],[]] => [[.,[[.,.],.]],.] => [2,3,1,4] => 2
[[[[]]],[]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 2
[[[],[],[]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[[],[[]]]] => [.,[[.,.],[.,.]]] => [4,2,3,1] => 1
[[[[]],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 1
[[[[],[]]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 1
[[[[[]]]]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[[],[],[],[],[]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 5
[[],[],[],[[]]] => [[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 4
[[],[],[[]],[]] => [[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 4
[[],[],[[],[]]] => [[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 3
[[],[],[[[]]]] => [[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 3
[[],[[]],[],[]] => [[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 4
[[],[[]],[[]]] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 3
[[],[[],[]],[]] => [[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 3
[[],[[[]]],[]] => [[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 3
[[],[[],[],[]]] => [[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 2
[[],[[],[[]]]] => [[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 2
[[],[[[]],[]]] => [[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 2
[[],[[[],[]]]] => [[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 2
[[],[[[[]]]]] => [[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 2
[[[]],[],[],[]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 4
[[[]],[],[[]]] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 3
[[[]],[[]],[]] => [[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 3
[[[]],[[],[]]] => [[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 2
[[[]],[[[]]]] => [[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 2
[[[],[]],[],[]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 3
[[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 3
[[[],[]],[[]]] => [[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 2
[[[[]]],[[]]] => [[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 2
[[[],[],[]],[]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 2
[[[],[[]]],[]] => [[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 2
[[[[]],[]],[]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 2
[[[[],[]]],[]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[[[[]]]],[]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 2
[[[],[],[],[]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[[],[],[[]]]] => [.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 1
[[[],[[]],[]]] => [.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 1
[[[],[[],[]]]] => [.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 1
[[[],[[[]]]]] => [.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 1
[[[[]],[],[]]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 1
[[[[]],[[]]]] => [.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 1
[[[[],[]],[]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 1
[[[[[]]],[]]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 1
[[[[],[],[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 1
[[[[],[[]]]]] => [.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 1
[[[[[]],[]]]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[[[[[],[]]]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 1
[[[[[[]]]]]] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1
[[],[],[],[],[],[]] => [[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 6
[[],[],[],[],[[]]] => [[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 5
[[],[],[],[[]],[]] => [[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 5
[[],[],[],[[],[]]] => [[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 4
[[],[],[],[[[]]]] => [[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 4
[[],[],[[]],[],[]] => [[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 5
[[],[],[[]],[[]]] => [[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 4
[[],[],[[],[]],[]] => [[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 4
[[],[],[[[]]],[]] => [[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 4
[[],[],[[],[],[]]] => [[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 3
[[],[],[[],[[]]]] => [[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 3
[[],[],[[[]],[]]] => [[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 3
[[],[],[[[],[]]]] => [[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 3
[[],[],[[[[]]]]] => [[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 3
[[],[[]],[],[],[]] => [[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 5
[[],[[]],[],[[]]] => [[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 4
[[],[[]],[[]],[]] => [[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 4
[[],[[]],[[],[]]] => [[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 3
[[],[[]],[[[]]]] => [[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 3
[[],[[],[]],[],[]] => [[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 4
[[],[[[]]],[],[]] => [[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 4
[[],[[],[]],[[]]] => [[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 3
[[],[[[]]],[[]]] => [[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 3
[[],[[],[],[]],[]] => [[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 3
[[],[[],[[]]],[]] => [[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 3
[[],[[[]],[]],[]] => [[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 3
[[],[[[],[]]],[]] => [[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 3
[[],[[[[]]]],[]] => [[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 3
[[],[[],[],[],[]]] => [[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 2
[[],[[],[],[[]]]] => [[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 2
[[],[[],[[]],[]]] => [[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 2
[[],[[],[[],[]]]] => [[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 2
[[],[[],[[[]]]]] => [[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 2
[[],[[[]],[],[]]] => [[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 2
[[],[[[]],[[]]]] => [[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 2
[[],[[[],[]],[]]] => [[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 2
[[],[[[[]]],[]]] => [[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 2
>>> Load all 196 entries. <<<
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searching the database for statistics with the same generating function
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..
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 $t$.
- 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 $t$.
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.
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.
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