Identifier
Values
[.,.] => [1] => [.,.] => [1,0] => 0
[.,[.,.]] => [2,1] => [[.,.],.] => [1,1,0,0] => 0
[[.,.],.] => [1,2] => [.,[.,.]] => [1,0,1,0] => 1
[.,[.,[.,.]]] => [3,2,1] => [[[.,.],.],.] => [1,1,1,0,0,0] => 0
[.,[[.,.],.]] => [2,3,1] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[[.,.],[.,.]] => [3,1,2] => [[.,[.,.]],.] => [1,1,0,1,0,0] => 2
[[.,[.,.]],.] => [2,1,3] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[.,[[.,.],[.,.]]] => [4,2,3,1] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 2
[.,[[.,[.,.]],.]] => [3,2,4,1] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[[.,.],[.,[.,.]]] => [4,3,1,2] => [[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => 3
[[.,[.,.]],[.,.]] => [4,2,1,3] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 2
[[.,[.,[.,.]]],.] => [3,2,1,4] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => 2
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => 3
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => 2
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => 4
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => 3
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => 2
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [[[[[.,.],.],[.,.]],.],.] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [[[[[.,.],[.,.]],.],.],.] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [[[[[.,.],.],[.,.]],.],.] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],.] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => [[[[[.,.],[.,.]],.],.],.] => [1,1,1,1,1,0,0,1,0,0,0,0] => 4
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => [[[[[.,.],.],[.,.]],.],.] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => [[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]] => [6,7,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]] => [7,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],.] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]] => [6,5,7,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]] => [7,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],.] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
[.,[.,[.,[[.,[.,.]],[.,.]]]]] => [7,5,4,6,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],.] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]] => [6,5,4,7,3,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]] => [7,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],.] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4
[.,[.,[[.,[.,.]],[.,[.,.]]]]] => [7,6,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],.] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
[.,[.,[[.,[.,[.,.]]],[.,.]]]] => [7,5,4,3,6,2,1] => [[[[[[.,.],.],.],.],[.,.]],.] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]] => [6,5,4,3,7,2,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]] => [7,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],.] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 5
[.,[[.,[.,.]],[.,[.,[.,.]]]]] => [7,6,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],.] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4
[.,[[.,[.,[.,.]]],[.,[.,.]]]] => [7,6,4,3,2,5,1] => [[[[[[.,.],.],.],[.,.]],.],.] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
[.,[[.,[.,[.,[.,.]]]],[.,.]]] => [7,5,4,3,2,6,1] => [[[[[[.,.],.],.],.],[.,.]],.] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[.,[[.,[.,[.,[.,[.,.]]]]],.]] => [6,5,4,3,2,7,1] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,.],[.,[.,[.,[.,[.,.]]]]]] => [7,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],.] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6
[[.,[.,.]],[.,[.,[.,[.,.]]]]] => [7,6,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],.] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 5
[[.,[.,[.,.]]],[.,[.,[.,.]]]] => [7,6,5,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],.] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 4
[[.,[.,[.,[.,.]]]],[.,[.,.]]] => [7,6,4,3,2,1,5] => [[[[[[.,.],.],.],[.,.]],.],.] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 3
[[.,[.,[.,[.,[.,.]]]]],[.,.]] => [7,5,4,3,2,1,6] => [[[[[[.,.],.],.],.],[.,.]],.] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 2
[[.,[.,[.,[.,[.,[.,.]]]]]],.] => [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 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 indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
  • a leaf is associated to the empty Dyck Word
  • a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
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.