Identifier
Values
[[]] => [.,.] => [1] => [1] => 1
[[],[]] => [.,[.,.]] => [2,1] => [2,1] => 2
[[[]]] => [[.,.],.] => [1,2] => [1,2] => 1
[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 6
[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => [2,3,1] => 3
[[[]],[]] => [[.,.],[.,.]] => [3,1,2] => [3,1,2] => 3
[[[],[]]] => [[.,[.,.]],.] => [2,1,3] => [2,1,3] => 2
[[[[]]]] => [[[.,.],.],.] => [1,2,3] => [1,2,3] => 1
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 24
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => 12
[[],[[]],[]] => [.,[[.,.],[.,.]]] => [4,2,3,1] => [4,2,3,1] => 12
[[],[[],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => 8
[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,4,1] => 4
[[[]],[],[]] => [[.,.],[.,[.,.]]] => [4,3,1,2] => [4,3,1,2] => 12
[[[]],[[]]] => [[.,.],[[.,.],.]] => [3,4,1,2] => [3,4,1,2] => 6
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [4,2,1,3] => [4,2,1,3] => 8
[[[[]]],[]] => [[[.,.],.],[.,.]] => [4,1,2,3] => [4,1,2,3] => 4
[[[],[],[]]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 6
[[[],[[]]]] => [[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1,4] => 3
[[[[]],[]]] => [[[.,.],[.,.]],.] => [3,1,2,4] => [3,1,2,4] => 3
[[[[],[]]]] => [[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 2
[[[[[]]]]] => [[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 1
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 1
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Description
The number of signed permutations less than or equal to a signed permutation in left weak order.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
to binary tree: right brother = right child
Description
Return a binary tree of size $n-1$ (where $n$ is the size of an ordered tree $t$) obtained from $t$ by the following recursive rule:
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left 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.