Identifier
Values
[.,.] => [[],[]] => [.,[.,.]] => [2,1] => 1
[.,[.,.]] => [[],[[],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 2
[[.,.],.] => [[[],[]],[]] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 2
[.,[.,[.,.]]] => [[],[[],[[],[]]]] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 1
[.,[[.,.],.]] => [[],[[[],[]],[]]] => [.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 3
[[.,.],[.,.]] => [[[],[]],[[],[]]] => [[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 3
[[.,[.,.]],.] => [[[],[[],[]]],[]] => [[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => 3
[[[.,.],.],.] => [[[[],[]],[]],[]] => [[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => 3
[.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]] => [5,4,6,3,7,2,8,1] => 2
[.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => [.,[[.,[[[.,[.,.]],[.,.]],.]],.]] => [4,3,6,5,7,2,8,1] => 2
[.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => [.,[[[.,[.,.]],[[.,[.,.]],.]],.]] => [3,2,6,5,7,4,8,1] => 2
[.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => [.,[[[.,[[.,[.,.]],.]],[.,.]],.]] => [4,3,5,2,7,6,8,1] => 2
[.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => [.,[[[[.,[.,.]],[.,.]],[.,.]],.]] => [3,2,5,4,7,6,8,1] => 4
[[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => [[.,[.,.]],[[.,[[.,[.,.]],.]],.]] => [2,1,6,5,7,4,8,3] => 2
[[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]] => [2,1,5,4,7,6,8,3] => 4
[[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]] => [3,2,4,1,7,6,8,5] => 4
[[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]] => [2,1,4,3,7,6,8,5] => 4
[[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => [[.,[[.,[[.,[.,.]],.]],.]],[.,.]] => [4,3,5,2,6,1,8,7] => 2
[[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => [[.,[[[.,[.,.]],[.,.]],.]],[.,.]] => [3,2,5,4,6,1,8,7] => 4
[[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]] => [2,1,5,4,6,3,8,7] => 4
[[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => [[[.,[[.,[.,.]],.]],[.,.]],[.,.]] => [3,2,4,1,6,5,8,7] => 4
[[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5,8,7] => 4
[[[[[.,.],.],.],.],.] => [[[[[[],[]],[]],[]],[]],[]] => [[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5,8,7,10,9] => 5
[[[[[[.,.],.],.],.],.],.] => [[[[[[[],[]],[]],[]],[]],[]],[]] => [[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of cycles in the cycle decomposition of a permutation.
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
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
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 $t$.