Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 2
[.,[.,[.,.]]] => [3,2,1] => 1
[.,[[.,.],.]] => [2,3,1] => 1
[[.,.],[.,.]] => [3,1,2] => 2
[[.,[.,.]],.] => [2,1,3] => 3
[[[.,.],.],.] => [1,2,3] => 3
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]] => [4,2,3,1] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]] => [4,3,1,2] => 2
[[.,.],[[.,.],.]] => [3,4,1,2] => 2
[[.,[.,.]],[.,.]] => [4,2,1,3] => 3
[[[.,.],.],[.,.]] => [4,1,2,3] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => 4
[[.,[[.,.],.]],.] => [2,3,1,4] => 4
[[[.,.],[.,.]],.] => [3,1,2,4] => 4
[[[.,[.,.]],.],.] => [2,1,3,4] => 4
[[[[.,.],.],.],.] => [1,2,3,4] => 4
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 1
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 1
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 1
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 2
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 3
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 3
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 3
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 4
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 4
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 4
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 4
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 5
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 5
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 5
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 5
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 5
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 5
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 5
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 5
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 5
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 5
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 5
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 5
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 5
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 1
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 1
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 1
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 1
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 1
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 1
>>> Load all 197 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 1
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 1
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 1
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 1
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 2
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 2
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 2
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 2
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 2
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 2
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 2
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 2
[[.,.],[[.,[[.,.],.]],.]] => [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] => 2
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 3
[[.,[.,.]],[.,[[.,.],.]]] => [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] => 3
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 3
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 3
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 3
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 3
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 3
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 4
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 4
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 4
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 4
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 4
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 4
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 4
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 4
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 4
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 5
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 5
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 5
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 5
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 5
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 5
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 5
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 5
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 5
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 5
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 5
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 5
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 5
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 6
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 6
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 6
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 6
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 6
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 6
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 6
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 6
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 6
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 6
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 6
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 6
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 6
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 6
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 6
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 6
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 6
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 6
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 6
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 6
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 6
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 6
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 6
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 6
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 6
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 6
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 6
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 6
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 6
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 6
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 6
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 6
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 6
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 6
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 6
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 6
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 6
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 6
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 6
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 6
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 6
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 6
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 7
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 last entry of a permutation.
This statistic is undefined for the empty 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.