Identifier
-
Mp00026:
Dyck paths
—to ordered tree⟶
Ordered trees
St000328: Ordered trees ⟶ ℤ
Values
[1,0] => [[]] => 1
[1,0,1,0] => [[],[]] => 2
[1,1,0,0] => [[[]]] => 1
[1,0,1,0,1,0] => [[],[],[]] => 3
[1,0,1,1,0,0] => [[],[[]]] => 2
[1,1,0,0,1,0] => [[[]],[]] => 2
[1,1,0,1,0,0] => [[[],[]]] => 2
[1,1,1,0,0,0] => [[[[]]]] => 1
[1,0,1,0,1,0,1,0] => [[],[],[],[]] => 4
[1,0,1,0,1,1,0,0] => [[],[],[[]]] => 3
[1,0,1,1,0,0,1,0] => [[],[[]],[]] => 3
[1,0,1,1,0,1,0,0] => [[],[[],[]]] => 2
[1,0,1,1,1,0,0,0] => [[],[[[]]]] => 2
[1,1,0,0,1,0,1,0] => [[[]],[],[]] => 3
[1,1,0,0,1,1,0,0] => [[[]],[[]]] => 2
[1,1,0,1,0,0,1,0] => [[[],[]],[]] => 2
[1,1,0,1,0,1,0,0] => [[[],[],[]]] => 3
[1,1,0,1,1,0,0,0] => [[[],[[]]]] => 2
[1,1,1,0,0,0,1,0] => [[[[]]],[]] => 2
[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] => [[],[],[],[],[]] => 5
[1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 4
[1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 4
[1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => 3
[1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 3
[1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 4
[1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 3
[1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => 3
[1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => 3
[1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => 2
[1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3
[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] => [[],[[[[]]]]] => 2
[1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 4
[1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 3
[1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 3
[1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => 2
[1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 2
[1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => 3
[1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => 2
[1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => 3
[1,1,0,1,0,1,0,1,0,0] => [[[],[],[],[]]] => 4
[1,1,0,1,0,1,1,0,0,0] => [[[],[],[[]]]] => 3
[1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => 2
[1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => 3
[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] => [[[[]]],[],[]] => 3
[1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 2
[1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => 2
[1,1,1,0,0,1,0,1,0,0] => [[[[]],[],[]]] => 3
[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] => [[[[],[],[]]]] => 3
[1,1,1,0,1,1,0,0,0,0] => [[[[],[[]]]]] => 2
[1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 2
[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] => [[],[],[],[],[],[]] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => 5
[1,0,1,0,1,0,1,1,0,1,0,0] => [[],[],[],[[],[]]] => 4
[1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [[],[],[[],[]],[]] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [[],[],[[],[],[]]] => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [[],[],[[],[[]]]] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [[],[],[[[]],[]]] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[]] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [[],[[]],[[],[]]] => 3
[1,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[[]]]] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [[],[[],[]],[],[]] => 4
[1,0,1,1,0,1,0,0,1,1,0,0] => [[],[[],[]],[[]]] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [[],[[],[],[]],[]] => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [[],[[],[],[],[]]] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [[],[[],[],[[]]]] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [[],[[],[[]]],[]] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [[],[[],[[]],[]]] => 3
[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] => [[],[[[]]],[],[]] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [[],[[[]]],[[]]] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [[],[[[]],[]],[]] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [[],[[[]],[],[]]] => 3
[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] => [[],[[[],[]]],[]] => 3
[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] => [[],[[[],[],[]]]] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => 2
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Description
The maximum number of child nodes in a tree.
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.
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.
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