Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001181: Dyck paths ⟶ ℤ
Values
[.,.] => [1] => [.,.] => [1,0] => 0
[.,[.,.]] => [2,1] => [[.,.],.] => [1,1,0,0] => 0
[[.,.],.] => [1,2] => [.,[.,.]] => [1,0,1,0] => 0
[.,[.,[.,.]]] => [3,2,1] => [[[.,.],.],.] => [1,1,1,0,0,0] => 0
[.,[[.,.],.]] => [2,3,1] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 0
[[.,.],[.,.]] => [1,3,2] => [.,[[.,.],.]] => [1,0,1,1,0,0] => 0
[[.,[.,.]],.] => [2,1,3] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 0
[[[.,.],.],.] => [1,2,3] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => 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] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 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] => 0
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 0
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => 0
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 1
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 1
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 1
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 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] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [[[.,.],.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [[[.,.],.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [[.,.],[[[[.,.],.],.],.]] => [1,1,0,0,1,1,1,1,0,0,0,0] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [[.,.],[[[.,.],.],[.,.]]] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [[.,.],[[.,.],[[.,.],.]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [[.,.],[[[.,.],.],[.,.]]] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [[.,.],[[.,.],[.,[.,.]]]] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [[.,.],[.,[[[.,.],.],.]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [[.,.],[.,[[.,.],[.,.]]]] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [[.,.],[[.,.],[[.,.],.]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [[.,.],[.,[.,[[.,.],.]]]] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0,1,0] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [[.,.],[[[.,.],.],[.,.]]] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [[.,.],[[.,.],[.,[.,.]]]] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [[.,.],[.,[[.,.],[.,.]]]] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
>>> Load all 196 entries. <<<
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.
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:
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 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.
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.
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