Processing math: 100%

Identifier
Values
[[]] => [[]] => 1
[[],[]] => [[[]]] => 1
[[[]]] => [[],[]] => 2
[[],[],[]] => [[[[]]]] => 1
[[],[[]]] => [[[],[]]] => 2
[[[]],[]] => [[],[[]]] => 2
[[[],[]]] => [[[]],[]] => 2
[[[[]]]] => [[],[],[]] => 3
[[],[],[],[]] => [[[[[]]]]] => 1
[[],[],[[]]] => [[[[],[]]]] => 2
[[],[[]],[]] => [[[],[[]]]] => 2
[[],[[],[]]] => [[[[]],[]]] => 2
[[],[[[]]]] => [[[],[],[]]] => 3
[[[]],[],[]] => [[],[[[]]]] => 2
[[[]],[[]]] => [[],[[],[]]] => 2
[[[],[]],[]] => [[[]],[[]]] => 2
[[[[]]],[]] => [[],[],[[]]] => 3
[[[],[],[]]] => [[[[]]],[]] => 2
[[[],[[]]]] => [[[],[]],[]] => 2
[[[[]],[]]] => [[],[[]],[]] => 3
[[[[],[]]]] => [[[]],[],[]] => 3
[[[[[]]]]] => [[],[],[],[]] => 4
[[],[],[],[],[]] => [[[[[[]]]]]] => 1
[[],[],[],[[]]] => [[[[[],[]]]]] => 2
[[],[],[[]],[]] => [[[[],[[]]]]] => 2
[[],[],[[],[]]] => [[[[[]],[]]]] => 2
[[],[],[[[]]]] => [[[[],[],[]]]] => 3
[[],[[]],[],[]] => [[[],[[[]]]]] => 2
[[],[[]],[[]]] => [[[],[[],[]]]] => 2
[[],[[],[]],[]] => [[[[]],[[]]]] => 2
[[],[[[]]],[]] => [[[],[],[[]]]] => 3
[[],[[],[],[]]] => [[[[[]]],[]]] => 2
[[],[[],[[]]]] => [[[[],[]],[]]] => 2
[[],[[[]],[]]] => [[[],[[]],[]]] => 3
[[],[[[],[]]]] => [[[[]],[],[]]] => 3
[[],[[[[]]]]] => [[[],[],[],[]]] => 4
[[[]],[],[],[]] => [[],[[[[]]]]] => 2
[[[]],[],[[]]] => [[],[[[],[]]]] => 2
[[[]],[[]],[]] => [[],[[],[[]]]] => 2
[[[]],[[],[]]] => [[],[[[]],[]]] => 2
[[[]],[[[]]]] => [[],[[],[],[]]] => 3
[[[],[]],[],[]] => [[[]],[[[]]]] => 2
[[[[]]],[],[]] => [[],[],[[[]]]] => 3
[[[],[]],[[]]] => [[[]],[[],[]]] => 2
[[[[]]],[[]]] => [[],[],[[],[]]] => 3
[[[],[],[]],[]] => [[[[]]],[[]]] => 2
[[[],[[]]],[]] => [[[],[]],[[]]] => 2
[[[[]],[]],[]] => [[],[[]],[[]]] => 3
[[[[],[]]],[]] => [[[]],[],[[]]] => 3
[[[[[]]]],[]] => [[],[],[],[[]]] => 4
[[[],[],[],[]]] => [[[[[]]]],[]] => 2
[[[],[],[[]]]] => [[[[],[]]],[]] => 2
[[[],[[]],[]]] => [[[],[[]]],[]] => 2
[[[],[[],[]]]] => [[[[]],[]],[]] => 2
[[[],[[[]]]]] => [[[],[],[]],[]] => 3
[[[[]],[],[]]] => [[],[[[]]],[]] => 3
[[[[]],[[]]]] => [[],[[],[]],[]] => 3
[[[[],[]],[]]] => [[[]],[[]],[]] => 3
[[[[[]]],[]]] => [[],[],[[]],[]] => 4
[[[[],[],[]]]] => [[[[]]],[],[]] => 3
[[[[],[[]]]]] => [[[],[]],[],[]] => 3
[[[[[]],[]]]] => [[],[[]],[],[]] => 4
[[[[[],[]]]]] => [[[]],[],[],[]] => 4
[[[[[[]]]]]] => [[],[],[],[],[]] => 5
[[],[],[],[],[],[]] => [[[[[[[]]]]]]] => 1
[[],[],[],[],[[]]] => [[[[[[],[]]]]]] => 2
[[],[],[],[[]],[]] => [[[[[],[[]]]]]] => 2
[[],[],[],[[],[]]] => [[[[[[]],[]]]]] => 2
[[],[],[],[[[]]]] => [[[[[],[],[]]]]] => 3
[[],[],[[]],[],[]] => [[[[],[[[]]]]]] => 2
[[],[],[[]],[[]]] => [[[[],[[],[]]]]] => 2
[[],[],[[],[]],[]] => [[[[[]],[[]]]]] => 2
[[],[],[[[]]],[]] => [[[[],[],[[]]]]] => 3
[[],[],[[],[],[]]] => [[[[[[]]],[]]]] => 2
[[],[],[[],[[]]]] => [[[[[],[]],[]]]] => 2
[[],[],[[[]],[]]] => [[[[],[[]],[]]]] => 3
[[],[],[[[],[]]]] => [[[[[]],[],[]]]] => 3
[[],[],[[[[]]]]] => [[[[],[],[],[]]]] => 4
[[],[[]],[],[],[]] => [[[],[[[[]]]]]] => 2
[[],[[]],[],[[]]] => [[[],[[[],[]]]]] => 2
[[],[[]],[[]],[]] => [[[],[[],[[]]]]] => 2
[[],[[]],[[],[]]] => [[[],[[[]],[]]]] => 2
[[],[[]],[[[]]]] => [[[],[[],[],[]]]] => 3
[[],[[],[]],[],[]] => [[[[]],[[[]]]]] => 2
[[],[[[]]],[],[]] => [[[],[],[[[]]]]] => 3
[[],[[],[]],[[]]] => [[[[]],[[],[]]]] => 2
[[],[[[]]],[[]]] => [[[],[],[[],[]]]] => 3
[[],[[],[],[]],[]] => [[[[[]]],[[]]]] => 2
[[],[[],[[]]],[]] => [[[[],[]],[[]]]] => 2
[[],[[[]],[]],[]] => [[[],[[]],[[]]]] => 3
[[],[[[],[]]],[]] => [[[[]],[],[[]]]] => 3
[[],[[[[]]]],[]] => [[[],[],[],[[]]]] => 4
[[],[[],[],[],[]]] => [[[[[[]]]],[]]] => 2
[[],[[],[],[[]]]] => [[[[[],[]]],[]]] => 2
[[],[[],[[]],[]]] => [[[[],[[]]],[]]] => 2
[[],[[],[[],[]]]] => [[[[[]],[]],[]]] => 2
[[],[[],[[[]]]]] => [[[[],[],[]],[]]] => 3
[[],[[[]],[],[]]] => [[[],[[[]]],[]]] => 3
[[],[[[]],[[]]]] => [[[],[[],[]],[]]] => 3
[[],[[[],[]],[]]] => [[[[]],[[]],[]]] => 3
[[],[[[[]]],[]]] => [[[],[],[[]],[]]] => 4
>>> Load all 196 entries. <<<
[[],[[[],[],[]]]] => [[[[[]]],[],[]]] => 3
[[],[[[],[[]]]]] => [[[[],[]],[],[]]] => 3
[[],[[[[]],[]]]] => [[[],[[]],[],[]]] => 4
[[],[[[[],[]]]]] => [[[[]],[],[],[]]] => 4
[[],[[[[[]]]]]] => [[[],[],[],[],[]]] => 5
[[[]],[],[],[],[]] => [[],[[[[[]]]]]] => 2
[[[]],[],[],[[]]] => [[],[[[[],[]]]]] => 2
[[[]],[],[[]],[]] => [[],[[[],[[]]]]] => 2
[[[]],[],[[],[]]] => [[],[[[[]],[]]]] => 2
[[[]],[],[[[]]]] => [[],[[[],[],[]]]] => 3
[[[]],[[]],[],[]] => [[],[[],[[[]]]]] => 2
[[[]],[[]],[[]]] => [[],[[],[[],[]]]] => 2
[[[]],[[],[]],[]] => [[],[[[]],[[]]]] => 2
[[[]],[[[]]],[]] => [[],[[],[],[[]]]] => 3
[[[]],[[],[],[]]] => [[],[[[[]]],[]]] => 2
[[[]],[[],[[]]]] => [[],[[[],[]],[]]] => 2
[[[]],[[[]],[]]] => [[],[[],[[]],[]]] => 3
[[[]],[[[],[]]]] => [[],[[[]],[],[]]] => 3
[[[]],[[[[]]]]] => [[],[[],[],[],[]]] => 4
[[[],[]],[],[],[]] => [[[]],[[[[]]]]] => 2
[[[[]]],[],[],[]] => [[],[],[[[[]]]]] => 3
[[[],[]],[],[[]]] => [[[]],[[[],[]]]] => 2
[[[[]]],[],[[]]] => [[],[],[[[],[]]]] => 3
[[[],[]],[[]],[]] => [[[]],[[],[[]]]] => 2
[[[[]]],[[]],[]] => [[],[],[[],[[]]]] => 3
[[[],[]],[[],[]]] => [[[]],[[[]],[]]] => 2
[[[],[]],[[[]]]] => [[[]],[[],[],[]]] => 3
[[[[]]],[[],[]]] => [[],[],[[[]],[]]] => 3
[[[[]]],[[[]]]] => [[],[],[[],[],[]]] => 3
[[[],[],[]],[],[]] => [[[[]]],[[[]]]] => 2
[[[],[[]]],[],[]] => [[[],[]],[[[]]]] => 2
[[[[]],[]],[],[]] => [[],[[]],[[[]]]] => 3
[[[[],[]]],[],[]] => [[[]],[],[[[]]]] => 3
[[[[[]]]],[],[]] => [[],[],[],[[[]]]] => 4
[[[],[],[]],[[]]] => [[[[]]],[[],[]]] => 2
[[[],[[]]],[[]]] => [[[],[]],[[],[]]] => 2
[[[[]],[]],[[]]] => [[],[[]],[[],[]]] => 3
[[[[],[]]],[[]]] => [[[]],[],[[],[]]] => 3
[[[[[]]]],[[]]] => [[],[],[],[[],[]]] => 4
[[[],[],[],[]],[]] => [[[[[]]]],[[]]] => 2
[[[],[],[[]]],[]] => [[[[],[]]],[[]]] => 2
[[[],[[]],[]],[]] => [[[],[[]]],[[]]] => 2
[[[],[[],[]]],[]] => [[[[]],[]],[[]]] => 2
[[[],[[[]]]],[]] => [[[],[],[]],[[]]] => 3
[[[[]],[],[]],[]] => [[],[[[]]],[[]]] => 3
[[[[]],[[]]],[]] => [[],[[],[]],[[]]] => 3
[[[[],[]],[]],[]] => [[[]],[[]],[[]]] => 3
[[[[[]]],[]],[]] => [[],[],[[]],[[]]] => 4
[[[[],[],[]]],[]] => [[[[]]],[],[[]]] => 3
[[[[],[[]]]],[]] => [[[],[]],[],[[]]] => 3
[[[[[]],[]]],[]] => [[],[[]],[],[[]]] => 4
[[[[[],[]]]],[]] => [[[]],[],[],[[]]] => 4
[[[[[[]]]]],[]] => [[],[],[],[],[[]]] => 5
[[[],[],[],[],[]]] => [[[[[[]]]]],[]] => 2
[[[],[],[],[[]]]] => [[[[[],[]]]],[]] => 2
[[[],[],[[]],[]]] => [[[[],[[]]]],[]] => 2
[[[],[],[[],[]]]] => [[[[[]],[]]],[]] => 2
[[[],[],[[[]]]]] => [[[[],[],[]]],[]] => 3
[[[],[[]],[],[]]] => [[[],[[[]]]],[]] => 2
[[[],[[]],[[]]]] => [[[],[[],[]]],[]] => 2
[[[],[[],[]],[]]] => [[[[]],[[]]],[]] => 2
[[[],[[[]]],[]]] => [[[],[],[[]]],[]] => 3
[[[],[[],[],[]]]] => [[[[[]]],[]],[]] => 2
[[[],[[],[[]]]]] => [[[[],[]],[]],[]] => 2
[[[],[[[]],[]]]] => [[[],[[]],[]],[]] => 3
[[[],[[[],[]]]]] => [[[[]],[],[]],[]] => 3
[[[],[[[[]]]]]] => [[[],[],[],[]],[]] => 4
[[[[]],[],[],[]]] => [[],[[[[]]]],[]] => 3
[[[[]],[],[[]]]] => [[],[[[],[]]],[]] => 3
[[[[]],[[]],[]]] => [[],[[],[[]]],[]] => 3
[[[[]],[[],[]]]] => [[],[[[]],[]],[]] => 3
[[[[]],[[[]]]]] => [[],[[],[],[]],[]] => 3
[[[[],[]],[],[]]] => [[[]],[[[]]],[]] => 3
[[[[[]]],[],[]]] => [[],[],[[[]]],[]] => 4
[[[[],[]],[[]]]] => [[[]],[[],[]],[]] => 3
[[[[[]]],[[]]]] => [[],[],[[],[]],[]] => 4
[[[[],[],[]],[]]] => [[[[]]],[[]],[]] => 3
[[[[],[[]]],[]]] => [[[],[]],[[]],[]] => 3
[[[[[]],[]],[]]] => [[],[[]],[[]],[]] => 4
[[[[[],[]]],[]]] => [[[]],[],[[]],[]] => 4
[[[[[[]]]],[]]] => [[],[],[],[[]],[]] => 5
[[[[],[],[],[]]]] => [[[[[]]]],[],[]] => 3
[[[[],[],[[]]]]] => [[[[],[]]],[],[]] => 3
[[[[],[[]],[]]]] => [[[],[[]]],[],[]] => 3
[[[[],[[],[]]]]] => [[[[]],[]],[],[]] => 3
[[[[],[[[]]]]]] => [[[],[],[]],[],[]] => 3
[[[[[]],[],[]]]] => [[],[[[]]],[],[]] => 4
[[[[[]],[[]]]]] => [[],[[],[]],[],[]] => 4
[[[[[],[]],[]]]] => [[[]],[[]],[],[]] => 4
[[[[[[]]],[]]]] => [[],[],[[]],[],[]] => 5
[[[[[],[],[]]]]] => [[[[]]],[],[],[]] => 4
[[[[[],[[]]]]]] => [[[],[]],[],[],[]] => 4
[[[[[[]],[]]]]] => [[],[[]],[],[],[]] => 5
[[[[[[],[]]]]]] => [[[]],[],[],[],[]] => 5
[[[[[[[]]]]]]] => [[],[],[],[],[],[]] => 6
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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
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Description
The maximum number of child nodes in a tree.
Map
DeBruijn-Morselt plane tree automorphism
Description
This automorphism on the set of plane trees with a given number of vertices is the combination of the two "canonical" bijections between plane trees and binary trees, using either the left or right branches. In [3] Shapiro essentially attributes this automorphism to DeBruijn and Morselt [1], hence the name I chose. Shapiro shows in that paper that for a large subset of plane trees (those corresponding to compositions), the sixth power of the automorphism acts as the identity on this subset. However, its behavior in general is quite chaotic. In [2] Donaghey studies this automorphism further and shows that its behavior on all trees can be reduced to its behavior on certain "primitive" trees.