- FindStat

Identifier

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000679: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

[1,1,0,1,0,0] => [[[],[]]] => 2

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

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

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

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

[1,0,1,1,0,1,0,0] => [[],[[],[]]] => 2

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

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

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

[1,1,0,1,0,0,1,0] => [[[],[]],[]] => 2

[1,1,0,1,0,1,0,0] => [[[],[],[]]] => 2

[1,1,0,1,1,0,0,0] => [[[],[[]]]] => 2

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

[1,1,1,0,0,1,0,0] => [[[[]],[]]] => 2

[1,1,1,0,1,0,0,0] => [[[[],[]]]] => 2

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

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

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

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

[1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => 2

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

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

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

[1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => 2

[1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => 2

[1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => 2

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

[1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => 2

[1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => 2

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

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

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

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

[1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => 2

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

[1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => 2

[1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => 2

[1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => 2

[1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => 2

[1,1,0,1,0,1,1,0,0,0] => [[[],[],[[]]]] => 2

[1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => 2

[1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => 2

[1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => 2

[1,1,0,1,1,1,0,0,0,0] => [[[],[[[]]]]] => 2

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

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

[1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => 2

[1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => 2

[1,1,1,0,0,1,1,0,0,0] => [[[[]],[[]]]] => 2

[1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => 2

[1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => 2

[1,1,1,0,1,0,1,0,0,0] => [[[[],[],[]]]] => 2

[1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 2

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

[1,1,1,1,0,0,0,1,0,0] => [[[[[]]],[]]] => 2

[1,1,1,1,0,0,1,0,0,0] => [[[[[]],[]]]] => 2

[1,1,1,1,0,1,0,0,0,0] => [[[[[],[]]]]] => 2

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

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

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

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

[1,0,1,0,1,0,1,1,0,1,0,0] => [[],[],[],[[],[]]] => 2

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

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

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [[],[],[[],[]],[]] => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => [[],[],[[],[],[]]] => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => [[],[],[[],[[]]]] => 2

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

[1,0,1,0,1,1,1,0,0,1,0,0] => [[],[],[[[]],[]]] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => 2

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

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

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

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

[1,0,1,1,0,0,1,1,0,1,0,0] => [[],[[]],[[],[]]] => 2

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

[1,0,1,1,0,1,0,0,1,0,1,0] => [[],[[],[]],[],[]] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [[],[[],[]],[[]]] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [[],[[],[],[]],[]] => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => [[],[[],[],[],[]]] => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => [[],[[],[],[[]]]] => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => [[],[[],[[]]],[]] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [[],[[],[[]],[]]] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [[],[[],[[],[]]]] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [[],[[],[[[]]]]] => 2

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

[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] => [[],[[[]],[]],[]] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => [[],[[[]],[],[]]] => 2

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

[1,0,1,1,1,0,1,0,1,0,0,0] => [[],[[[],[],[]]]] => 2

[1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => 2

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Description

The pruning number of an ordered tree.
A hanging branch of an ordered tree is a proper factor of the form $[^r]^r$ for some $r\geq 1$. A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch.
A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to $[]$.

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.