Identifier
-
Mp00139:
Ordered trees
—Zeilberger's Strahler bijection⟶
Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001174: Permutations ⟶ ℤ
Values
[[],[]] => [.,[.,.]] => [2,1] => 0
[[[]]] => [[.,.],.] => [1,2] => 0
[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => 0
[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => 1
[[[]],[]] => [[.,[.,.]],.] => [2,1,3] => 0
[[[],[]]] => [[.,.],[.,.]] => [1,3,2] => 0
[[[[]]]] => [[[.,.],.],.] => [1,2,3] => 0
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 0
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 1
[[],[[]],[]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 1
[[],[[],[]]] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 1
[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[[]],[],[]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 0
[[[]],[[]]] => [[.,[[.,.],.]],.] => [2,3,1,4] => 1
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [2,1,4,3] => 0
[[[[]]],[]] => [[[.,[.,.]],.],.] => [2,1,3,4] => 0
[[[],[],[]]] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 0
[[[],[[]]]] => [[.,.],[[.,.],.]] => [1,3,4,2] => 1
[[[[]],[]]] => [[[.,.],.],[.,.]] => [1,2,4,3] => 0
[[[[],[]]]] => [[[.,.],[.,.]],.] => [1,3,2,4] => 0
[[[[[]]]]] => [[[[.,.],.],.],.] => [1,2,3,4] => 0
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 1
[[],[],[[]],[]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[[],[],[[],[]]] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 1
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 1
[[],[[]],[],[]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 1
[[],[[]],[[]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 1
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 1
[[],[[[]]],[]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 1
[[],[[],[],[]]] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 1
[[],[[],[[]]]] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 1
[[],[[[]],[]]] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 1
[[],[[[],[]]]] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 1
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[[]],[],[],[]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 0
[[[]],[],[[]]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 1
[[[]],[[]],[]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 1
[[[]],[[],[]]] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 1
[[[]],[[[]]]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 1
[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 0
[[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 0
[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 1
[[[[]]],[[]]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 1
[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 0
[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 1
[[[[]],[]],[]] => [[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 0
[[[[],[]]],[]] => [[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 0
[[[[[]]]],[]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 0
[[[],[],[],[]]] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[[[],[],[[]]]] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 1
[[[],[[]],[]]] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 1
[[[],[[],[]]]] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 1
[[[],[[[]]]]] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 1
[[[[]],[],[]]] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 0
[[[[]],[[]]]] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 1
[[[[],[]],[]]] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 0
[[[[[]]],[]]] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 0
[[[[],[],[]]]] => [[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 0
[[[[],[[]]]]] => [[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 1
[[[[[]],[]]]] => [[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 0
[[[[[],[]]]]] => [[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 0
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 0
[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 1
[[],[],[],[[]],[]] => [.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 1
[[],[],[],[[],[]]] => [.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 1
[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 1
[[],[],[[]],[],[]] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 1
[[],[],[[]],[[]]] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 1
[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 1
[[],[],[[[]]],[]] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 1
[[],[],[[],[],[]]] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 1
[[],[],[[],[[]]]] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 1
[[],[],[[[]],[]]] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 1
[[],[],[[[],[]]]] => [.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 1
[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 1
[[],[[]],[],[],[]] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 1
[[],[[]],[],[[]]] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 1
[[],[[]],[[]],[]] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 1
[[],[[]],[[],[]]] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 1
[[],[[]],[[[]]]] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 1
[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 1
[[],[[[]]],[],[]] => [.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 1
[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 1
[[],[[[]]],[[]]] => [.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 1
[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 1
[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 1
[[],[[[]],[]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 1
[[],[[[],[]]],[]] => [.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 1
[[],[[[[]]]],[]] => [.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 1
[[],[[],[],[],[]]] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 1
[[],[[],[],[[]]]] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 1
[[],[[],[[]],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 1
[[],[[],[[],[]]]] => [.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 1
[[],[[],[[[]]]]] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 1
[[],[[[]],[],[]]] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 1
[[],[[[]],[[]]]] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 1
[[],[[[],[]],[]]] => [.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 1
[[],[[[[]]],[]]] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 1
[[],[[[],[],[]]]] => [.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 1
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Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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 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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