Identifier
Values
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => 2
[.,[[.,.],.]] => [2,3,1] => 1
[[.,.],[.,.]] => [3,1,2] => 1
[[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 3
[.,[.,[[.,.],.]]] => [3,4,2,1] => 2
[.,[[.,.],[.,.]]] => [4,2,3,1] => 2
[.,[[.,[.,.]],.]] => [3,2,4,1] => 2
[.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]] => [4,3,1,2] => 2
[[.,.],[[.,.],.]] => [3,4,1,2] => 1
[[.,[.,.]],[.,.]] => [4,2,1,3] => 2
[[[.,.],.],[.,.]] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[[.,.],[.,.]],.] => [3,1,2,4] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 4
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 3
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 3
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 2
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 3
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 3
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 2
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 3
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 2
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 2
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 3
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 3
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 3
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 2
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 2
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 3
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 2
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 2
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 1
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 2
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 1
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 5
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 4
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 4
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 4
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 4
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 4
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 4
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 3
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 3
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 3
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 2
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 4
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 3
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 3
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 3
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 2
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 4
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 3
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 3
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 2
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 4
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 3
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 3
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 3
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 2
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 4
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 3
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 3
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 3
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 2
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 3
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 2
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 3
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 2
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 3
>>> Load all 196 entries. <<<
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 2
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 2
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 2
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 4
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 3
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 3
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 3
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 2
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 3
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 3
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 3
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 2
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 2
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 2
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 4
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 3
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 3
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 3
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 2
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 3
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 2
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 2
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 2
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 4
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 3
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 3
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 2
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 3
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 2
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 3
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 2
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 2
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 4
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 3
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 3
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 3
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 2
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 3
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 2
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 3
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 2
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 3
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 2
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 2
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 2
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 4
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 3
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 3
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 3
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 2
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 3
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 2
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 3
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 3
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 2
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 2
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 2
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 3
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 2
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 2
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 2
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 1
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 3
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 2
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 2
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 1
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 3
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 2
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 2
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 2
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 3
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 2
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 2
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 1
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 2
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 1
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 2
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 1
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 2
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 1
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 1
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 1
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 0
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 0
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 indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
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.