- FindStat

Identifier

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees

Images

[1,0] => [[]] => [[]]

[1,0,1,0] => [[],[]] => [[[]]]

[1,1,0,0] => [[[]]] => [[],[]]

[1,0,1,0,1,0] => [[],[],[]] => [[[[]]]]

[1,0,1,1,0,0] => [[],[[]]] => [[[],[]]]

[1,1,0,0,1,0] => [[[]],[]] => [[],[[]]]

[1,1,0,1,0,0] => [[[],[]]] => [[[]],[]]

[1,1,1,0,0,0] => [[[[]]]] => [[],[],[]]

[1,0,1,0,1,0,1,0] => [[],[],[],[]] => [[[[[]]]]]

[1,0,1,0,1,1,0,0] => [[],[],[[]]] => [[[[],[]]]]

[1,0,1,1,0,0,1,0] => [[],[[]],[]] => [[[],[[]]]]

[1,0,1,1,0,1,0,0] => [[],[[],[]]] => [[[[]],[]]]

[1,0,1,1,1,0,0,0] => [[],[[[]]]] => [[[],[],[]]]

[1,1,0,0,1,0,1,0] => [[[]],[],[]] => [[],[[[]]]]

[1,1,0,0,1,1,0,0] => [[[]],[[]]] => [[],[[],[]]]

[1,1,0,1,0,0,1,0] => [[[],[]],[]] => [[[]],[[]]]

[1,1,0,1,0,1,0,0] => [[[],[],[]]] => [[[[]]],[]]

[1,1,0,1,1,0,0,0] => [[[],[[]]]] => [[[],[]],[]]

[1,1,1,0,0,0,1,0] => [[[[]]],[]] => [[],[],[[]]]

[1,1,1,0,0,1,0,0] => [[[[]],[]]] => [[],[[]],[]]

[1,1,1,0,1,0,0,0] => [[[[],[]]]] => [[[]],[],[]]

[1,1,1,1,0,0,0,0] => [[[[[]]]]] => [[],[],[],[]]

[1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => [[[[[[]]]]]]

[1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => [[[[[],[]]]]]

[1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => [[[[],[[]]]]]

[1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => [[[[[]],[]]]]

[1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => [[[[],[],[]]]]

[1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => [[[],[[[]]]]]

[1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => [[[],[[],[]]]]

[1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => [[[[]],[[]]]]

[1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => [[[[[]]],[]]]

[1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => [[[[],[]],[]]]

[1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => [[[],[],[[]]]]

[1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => [[[],[[]],[]]]

[1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => [[[[]],[],[]]]

[1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => [[[],[],[],[]]]

[1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => [[],[[[[]]]]]

[1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => [[],[[[],[]]]]

[1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => [[],[[],[[]]]]

[1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => [[],[[[]],[]]]

[1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => [[],[[],[],[]]]

[1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => [[[]],[[[]]]]

[1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => [[[]],[[],[]]]

[1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => [[[[]]],[[]]]

[1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => [[[[[]]]],[]]

[1,1,0,1,0,1,1,0,0,0] => [[[],[],[[]]]] => [[[[],[]]],[]]

[1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => [[[],[]],[[]]]

[1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => [[[],[[]]],[]]

[1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => [[[[]],[]],[]]

[1,1,0,1,1,1,0,0,0,0] => [[[],[[[]]]]] => [[[],[],[]],[]]

[1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => [[],[],[[[]]]]

[1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => [[],[],[[],[]]]

[1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => [[],[[]],[[]]]

[1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => [[],[[[]]],[]]

[1,1,1,0,0,1,1,0,0,0] => [[[[]],[[]]]] => [[],[[],[]],[]]

[1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => [[[]],[],[[]]]

[1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => [[[]],[[]],[]]

[1,1,1,0,1,0,1,0,0,0] => [[[[],[],[]]]] => [[[[]]],[],[]]

[1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => [[[],[]],[],[]]

[1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => [[],[],[],[[]]]

[1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => [[],[],[[]],[]]

[1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => [[],[[]],[],[]]

[1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => [[[]],[],[],[]]

[1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => [[],[],[],[],[]]

[1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[]] => [[[[[[[]]]]]]]

[1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => [[[[[[],[]]]]]]

[1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => [[[[[],[[]]]]]]

[1,0,1,0,1,0,1,1,0,1,0,0] => [[],[],[],[[],[]]] => [[[[[[]],[]]]]]

[1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => [[[[[],[],[]]]]]

[1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => [[[[],[[[]]]]]]

[1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => [[[[],[[],[]]]]]

[1,0,1,0,1,1,0,1,0,0,1,0] => [[],[],[[],[]],[]] => [[[[[]],[[]]]]]

[1,0,1,0,1,1,0,1,0,1,0,0] => [[],[],[[],[],[]]] => [[[[[[]]],[]]]]

[1,0,1,0,1,1,0,1,1,0,0,0] => [[],[],[[],[[]]]] => [[[[[],[]],[]]]]

[1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => [[[[],[],[[]]]]]

[1,0,1,0,1,1,1,0,0,1,0,0] => [[],[],[[[]],[]]] => [[[[],[[]],[]]]]

[1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => [[[[[]],[],[]]]]

[1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => [[[[],[],[],[]]]]

[1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => [[[],[[[[]]]]]]

[1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => [[[],[[[],[]]]]]

[1,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[]] => [[[],[[],[[]]]]]

[1,0,1,1,0,0,1,1,0,1,0,0] => [[],[[]],[[],[]]] => [[[],[[[]],[]]]]

[1,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[[]]]] => [[[],[[],[],[]]]]

[1,0,1,1,0,1,0,0,1,0,1,0] => [[],[[],[]],[],[]] => [[[[]],[[[]]]]]

[1,0,1,1,0,1,0,0,1,1,0,0] => [[],[[],[]],[[]]] => [[[[]],[[],[]]]]

[1,0,1,1,0,1,0,1,0,0,1,0] => [[],[[],[],[]],[]] => [[[[[]]],[[]]]]

[1,0,1,1,0,1,0,1,0,1,0,0] => [[],[[],[],[],[]]] => [[[[[[]]]],[]]]

[1,0,1,1,0,1,0,1,1,0,0,0] => [[],[[],[],[[]]]] => [[[[[],[]]],[]]]

[1,0,1,1,0,1,1,0,0,0,1,0] => [[],[[],[[]]],[]] => [[[[],[]],[[]]]]

[1,0,1,1,0,1,1,0,0,1,0,0] => [[],[[],[[]],[]]] => [[[[],[[]]],[]]]

[1,0,1,1,0,1,1,0,1,0,0,0] => [[],[[],[[],[]]]] => [[[[[]],[]],[]]]

[1,0,1,1,0,1,1,1,0,0,0,0] => [[],[[],[[[]]]]] => [[[[],[],[]],[]]]

[1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => [[[],[],[[[]]]]]

[1,0,1,1,1,0,0,0,1,1,0,0] => [[],[[[]]],[[]]] => [[[],[],[[],[]]]]

[1,0,1,1,1,0,0,1,0,0,1,0] => [[],[[[]],[]],[]] => [[[],[[]],[[]]]]

[1,0,1,1,1,0,0,1,0,1,0,0] => [[],[[[]],[],[]]] => [[[],[[[]]],[]]]

[1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => [[[],[[],[]],[]]]

[1,0,1,1,1,0,1,0,0,0,1,0] => [[],[[[],[]]],[]] => [[[[]],[],[[]]]]

[1,0,1,1,1,0,1,0,0,1,0,0] => [[],[[[],[]],[]]] => [[[[]],[[]],[]]]

[1,0,1,1,1,0,1,0,1,0,0,0] => [[],[[[],[],[]]]] => [[[[[]]],[],[]]]

[1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => [[[[],[]],[],[]]]

>>> Load all 2056 entries. <<<

Map

to ordered tree

Description

Sends a Dyck path to the ordered tree encoding the heights of the path.
This map is recursively defined as follows: A Dyck path $D$ of semilength $n$ may be decomposed, according to its returns (St000011The number of touch points (or returns) of a Dyck path.), into smaller paths $D_1,\dots,D_k$ of respective semilengths $n_1,\dots,n_k$ (so one has $n = n_1 + \dots n_k$) each of which has no returns.
Denote by $\tilde D_i$ the path of semilength $n_i-1$ obtained from $D_i$ by removing the initial up- and the final down-step.
This map then sends $D$ to the tree $T$ having a root note with ordered children $T_1,\dots,T_k$ which are again ordered trees computed from $D_1,\dots,D_k$ respectively.
The unique path of semilength $1$ is sent to the tree consisting of a single node.

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.