Identifier
-
Mp00139:
Ordered trees
—Zeilberger's Strahler bijection⟶
Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001294: Dyck paths ⟶ ℤ
Values
[[]] => [.,.] => [1,0] => 0
[[],[]] => [.,[.,.]] => [1,0,1,0] => 1
[[[]]] => [[.,.],.] => [1,1,0,0] => 0
[[],[],[]] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => 2
[[],[[]]] => [.,[[.,.],.]] => [1,0,1,1,0,0] => 1
[[[]],[]] => [[.,[.,.]],.] => [1,1,0,1,0,0] => 1
[[[],[]]] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[[[[]]]] => [[[.,.],.],.] => [1,1,1,0,0,0] => 0
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => 3
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => 2
[[],[[]],[]] => [.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => 2
[[],[[],[]]] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => 1
[[],[[[]]]] => [.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => 1
[[[]],[],[]] => [[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => 2
[[[]],[[]]] => [[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => 1
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => 1
[[[[]]],[]] => [[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => 1
[[[],[],[]]] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => 2
[[[],[[]]]] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => 1
[[[[]],[]]] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[[[[],[]]]] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => 1
[[[[[]]]]] => [[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 0
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => 4
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => 3
[[],[],[[]],[]] => [.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => 3
[[],[],[[],[]]] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => 2
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => 2
[[],[[]],[],[]] => [.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => 3
[[],[[]],[[]]] => [.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => 2
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => 2
[[],[[[]]],[]] => [.,[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,1,0,0,0] => 2
[[],[[],[],[]]] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => 2
[[],[[],[[]]]] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => 1
[[],[[[]],[]]] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => 1
[[],[[[],[]]]] => [.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 1
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[]],[],[],[]] => [[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => 2
[[[]],[],[[]]] => [[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => 2
[[[]],[[]],[]] => [[.,[[.,[.,.]],.]],.] => [1,1,0,1,1,0,1,0,0,0] => 2
[[[]],[[],[]]] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => 1
[[[]],[[[]]]] => [[.,[[[.,.],.],.]],.] => [1,1,0,1,1,1,0,0,0,0] => 1
[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => 2
[[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.] => [1,1,1,0,1,0,1,0,0,0] => 2
[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => 1
[[[[]]],[[]]] => [[[.,[[.,.],.]],.],.] => [1,1,1,0,1,1,0,0,0,0] => 1
[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => 3
[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => 1
[[[[]],[]],[]] => [[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => 1
[[[[],[]]],[]] => [[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => 1
[[[[[]]]],[]] => [[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => 1
[[[],[],[],[]]] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => 3
[[[],[],[[]]]] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => 2
[[[],[[]],[]]] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => 2
[[[],[[],[]]]] => [[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => 1
[[[],[[[]]]]] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => 1
[[[[]],[],[]]] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => 2
[[[[]],[[]]]] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => 1
[[[[],[]],[]]] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => 1
[[[[[]]],[]]] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[[[],[],[]]]] => [[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => 2
[[[[],[[]]]]] => [[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => 1
[[[[[]],[]]]] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => 1
[[[[[],[]]]]] => [[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => 1
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => 0
[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 5
[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
[[],[],[],[[]],[]] => [.,[.,[.,[[.,[.,.]],.]]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 4
[[],[],[],[[],[]]] => [.,[.,[.,[[.,.],[.,.]]]]] => [1,0,1,0,1,0,1,1,0,0,1,0] => 3
[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[[],[],[[]],[],[]] => [.,[.,[[.,[.,[.,.]]],.]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 4
[[],[],[[]],[[]]] => [.,[.,[[.,[[.,.],.]],.]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 3
[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 3
[[],[],[[[]]],[]] => [.,[.,[[[.,[.,.]],.],.]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
[[],[],[[],[],[]]] => [.,[.,[[.,.],[.,[.,.]]]]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[[],[],[[],[[]]]] => [.,[.,[[.,.],[[.,.],.]]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[[],[],[[[]],[]]] => [.,[.,[[[.,.],.],[.,.]]]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2
[[],[],[[[],[]]]] => [.,[.,[[[.,.],[.,.]],.]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 2
[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[[],[[]],[],[],[]] => [.,[[.,[.,[.,[.,.]]]],.]] => [1,0,1,1,0,1,0,1,0,1,0,0] => 3
[[],[[]],[],[[]]] => [.,[[.,[.,[[.,.],.]]],.]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 3
[[],[[]],[[]],[]] => [.,[[.,[[.,[.,.]],.]],.]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 3
[[],[[]],[[],[]]] => [.,[[.,.],[[.,.],[.,.]]]] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
[[],[[]],[[[]]]] => [.,[[.,[[[.,.],.],.]],.]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 2
[[],[[[]]],[],[]] => [.,[[[.,[.,[.,.]]],.],.]] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 2
[[],[[[]]],[[]]] => [.,[[[.,[[.,.],.]],.],.]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 4
[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 2
[[],[[[]],[]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[[],[[[],[]]],[]] => [.,[[[.,[.,.]],[.,.]],.]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 2
[[],[[[[]]]],[]] => [.,[[[[.,[.,.]],.],.],.]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[[],[[],[],[],[]]] => [.,[[.,.],[.,[.,[.,.]]]]] => [1,0,1,1,0,0,1,0,1,0,1,0] => 3
[[],[[],[],[[]]]] => [.,[[.,.],[.,[[.,.],.]]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
[[],[[],[[]],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => 2
[[],[[],[[],[]]]] => [.,[[.,[[.,.],[.,.]]],.]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 2
[[],[[],[[[]]]]] => [.,[[.,.],[[[.,.],.],.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1
[[],[[[]],[],[]]] => [.,[[[.,.],.],[.,[.,.]]]] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
[[],[[[]],[[]]]] => [.,[[[.,.],.],[[.,.],.]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1
[[],[[[],[]],[]]] => [.,[[[.,.],[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 1
[[],[[[[]]],[]]] => [.,[[[[.,.],.],.],[.,.]]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
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Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras.
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers oeis.org/A001006.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras.
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers oeis.org/A001006.
Map
Zeilberger's Strahler bijection
Description
Zeilberger's Strahler bijection between ordered and binary trees.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the binary tree.
This is a bijection sending the pruning number of the ordered tree to the Strahler number of the 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 $r$.
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