Processing math: 100%

Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => 1
[.,[[.,.],.]] => [2,3,1] => 1
[[.,.],[.,.]] => [3,1,2] => 1
[[.,[.,.]],.] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => 1
[.,[.,[.,[.,.]]]] => [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] => 1
[[.,.],[[.,.],.]] => [3,4,1,2] => 1
[[.,[.,.]],[.,.]] => [4,2,1,3] => 1
[[[.,.],.],[.,.]] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[[.,.],[.,.]],.] => [3,1,2,4] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]] => [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] => 1
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 1
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 1
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 1
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 1
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 1
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 1
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 1
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 1
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 1
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 1
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [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 236 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] => 1
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 1
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 1
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 1
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 1
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 1
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 1
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 1
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 1
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 1
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 1
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 1
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 1
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 1
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 1
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 1
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 1
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 1
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 1
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 1
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 1
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 1
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 1
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 1
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 1
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 1
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 1
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 1
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 1
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 1
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 1
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 1
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 1
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 1
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 1
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 1
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 1
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 1
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 1
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 1
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 1
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 1
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 1
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 1
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 1
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 1
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 1
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 1
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 1
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 1
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 1
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 1
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 1
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 1
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 1
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 1
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 1
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 1
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 1
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 1
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 1
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 1
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 1
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 1
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 1
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 1
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 1
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 1
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 1
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 1
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 1
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 1
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 1
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 1
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 1
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 1
[[[[.,[[.,.],.]],.],.],.] => [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] => 1
[.,[[.,[[[[.,.],.],.],.]],.]] => [3,4,5,6,2,7,1] => 1
[.,[[[[[.,.],.],.],[.,.]],.]] => [6,2,3,4,5,7,1] => 1
[.,[[[.,[[.,[.,.]],.]],.],.]] => [4,3,5,2,6,7,1] => 1
[.,[[[.,[[[.,.],.],.]],.],.]] => [3,4,5,2,6,7,1] => 1
[.,[[[[.,.],[[.,.],.]],.],.]] => [4,5,2,3,6,7,1] => 1
[.,[[[[.,[.,.]],[.,.]],.],.]] => [5,3,2,4,6,7,1] => 1
[.,[[[[[.,.],.],[.,.]],.],.]] => [5,2,3,4,6,7,1] => 1
[.,[[[[.,[.,[.,.]]],.],.],.]] => [4,3,2,5,6,7,1] => 1
[.,[[[[.,[[.,.],.]],.],.],.]] => [3,4,2,5,6,7,1] => 1
[.,[[[[[.,.],[.,.]],.],.],.]] => [4,2,3,5,6,7,1] => 1
[.,[[[[[.,[.,.]],.],.],.],.]] => [3,2,4,5,6,7,1] => 1
[.,[[[[[[.,.],.],.],.],.],.]] => [2,3,4,5,6,7,1] => 1
[[.,.],[[[[[.,.],.],.],.],.]] => [3,4,5,6,7,1,2] => 1
[[[[.,.],.],.],[[[.,.],.],.]] => [5,6,7,1,2,3,4] => 1
[[.,[.,[.,[.,[[.,.],.]]]]],.] => [5,6,4,3,2,1,7] => 1
[[.,[.,[.,[[.,[.,.]],.]]]],.] => [5,4,6,3,2,1,7] => 1
[[.,[.,[.,[[[.,.],.],.]]]],.] => [4,5,6,3,2,1,7] => 1
[[.,[.,[[[.,.],.],[.,.]]]],.] => [6,3,4,5,2,1,7] => 1
[[.,[.,[[.,[.,[.,.]]],.]]],.] => [5,4,3,6,2,1,7] => 1
[[.,[.,[[.,[[.,.],.]],.]]],.] => [4,5,3,6,2,1,7] => 1
[[.,[[.,[.,[.,.]]],[.,.]]],.] => [6,4,3,2,5,1,7] => 1
[[.,[[.,[[.,.],.]],[.,.]]],.] => [6,3,4,2,5,1,7] => 1
[[.,[[.,[.,[.,[.,.]]]],.]],.] => [5,4,3,2,6,1,7] => 1
[[.,[[[[[.,.],.],.],.],.]],.] => [2,3,4,5,6,1,7] => 1
[[[.,.],[[.,.],[[.,.],.]]],.] => [5,6,3,4,1,2,7] => 1
[[[[.,.],.],[[[.,.],.],.]],.] => [4,5,6,1,2,3,7] => 1
[[[[[.,.],.],.],[[.,.],.]],.] => [5,6,1,2,3,4,7] => 1
[[[[[[.,.],.],.],.],[.,.]],.] => [6,1,2,3,4,5,7] => 1
[[[.,[.,[.,[.,[.,.]]]]],.],.] => [5,4,3,2,1,6,7] => 1
[[[.,[[[[.,.],.],.],.]],.],.] => [2,3,4,5,1,6,7] => 1
[[[[[[.,.],.],.],[.,.]],.],.] => [5,1,2,3,4,6,7] => 1
[[[[.,[.,[.,[.,.]]]],.],.],.] => [4,3,2,1,5,6,7] => 1
[[[[.,[[[.,.],.],.]],.],.],.] => [2,3,4,1,5,6,7] => 1
[[[[[.,.],[[.,.],.]],.],.],.] => [3,4,1,2,5,6,7] => 1
[[[[[[.,.],.],[.,.]],.],.],.] => [4,1,2,3,5,6,7] => 1
[[[[[.,[.,[.,.]]],.],.],.],.] => [3,2,1,4,5,6,7] => 1
[[[[[.,[[.,.],.]],.],.],.],.] => [2,3,1,4,5,6,7] => 1
[[[[[[.,.],[.,.]],.],.],.],.] => [3,1,2,4,5,6,7] => 1
[[[[[[.,[.,.]],.],.],.],.],.] => [2,1,3,4,5,6,7] => 1
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 1
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
Description
The number of reduced Kogan faces with the permutation as type.
This is equivalent to finding the number of ways to represent the permutation π∈Sn+1 as a reduced subword of sn(sn−1sn)(sn−2sn−1sn)⋯(s1⋯sn), or the number of reduced pipe dreams for π.
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.