Identifier
Values
[[],[]] => [.,[.,.]] => [2,1] => 1
[[[]]] => [[.,.],.] => [1,2] => 0
[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => 1
[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => 1
[[[]],[]] => [[.,[.,.]],.] => [2,1,3] => 1
[[[],[]]] => [[.,.],[.,.]] => [3,1,2] => 1
[[[[]]]] => [[[.,.],.],.] => [1,2,3] => 0
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 2
[[],[[]],[]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 2
[[],[[],[]]] => [.,[[.,.],[.,.]]] => [4,2,3,1] => 1
[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[[]],[],[]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 2
[[[]],[[]]] => [[.,[[.,.],.]],.] => [2,3,1,4] => 2
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [4,2,1,3] => 2
[[[[]]],[]] => [[[.,[.,.]],.],.] => [2,1,3,4] => 1
[[[],[],[]]] => [[.,.],[.,[.,.]]] => [4,3,1,2] => 2
[[[],[[]]]] => [[.,.],[[.,.],.]] => [3,4,1,2] => 2
[[[[]],[]]] => [[[.,.],.],[.,.]] => [4,1,2,3] => 1
[[[[],[]]]] => [[[.,.],[.,.]],.] => [3,1,2,4] => 2
[[[[[]]]]] => [[[[.,.],.],.],.] => [1,2,3,4] => 0
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 2
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 2
[[],[],[[]],[]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 2
[[],[],[[],[]]] => [.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 2
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 2
[[],[[]],[],[]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 2
[[],[[]],[[]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 2
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 1
[[],[[[]]],[]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 2
[[],[[],[],[]]] => [.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 2
[[],[[],[[]]]] => [.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 2
[[],[[[]],[]]] => [.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 1
[[],[[[],[]]]] => [.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 2
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[[]],[],[],[]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 2
[[[]],[],[[]]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[[]],[[]],[]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 2
[[[]],[[],[]]] => [[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 2
[[[]],[[[]]]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 2
[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 2
[[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 2
[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 2
[[[[]]],[[]]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 2
[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 2
[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 2
[[[[]],[]],[]] => [[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 2
[[[[],[]]],[]] => [[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 2
[[[[[]]]],[]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1
[[[],[],[],[]]] => [[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 2
[[[],[],[[]]]] => [[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 2
[[[],[[]],[]]] => [[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 2
[[[],[[],[]]]] => [[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 2
[[[],[[[]]]]] => [[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 2
[[[[]],[],[]]] => [[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 2
[[[[]],[[]]]] => [[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 2
[[[[],[]],[]]] => [[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 2
[[[[[]]],[]]] => [[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 1
[[[[],[],[]]]] => [[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 2
[[[[],[[]]]]] => [[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 2
[[[[[]],[]]]] => [[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 2
[[[[[],[]]]]] => [[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 2
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
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Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
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.
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.