Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤ
Values
[1] => [1,0] => [[]] => 1
[1,1] => [1,0,1,0] => [[],[]] => 1
[2] => [1,1,0,0] => [[[]]] => 2
[1,1,1] => [1,0,1,0,1,0] => [[],[],[]] => 1
[1,2] => [1,0,1,1,0,0] => [[],[[]]] => 1
[2,1] => [1,1,0,0,1,0] => [[[]],[]] => 1
[3] => [1,1,1,0,0,0] => [[[[]]]] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => 1
[1,2,1] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => 1
[1,3] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => 1
[2,2] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => 2
[3,1] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 1
[4] => [1,1,1,1,0,0,0,0] => [[[[[]]]]] => 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[]] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 1
[5] => [1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[]] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[[]]] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[[]],[]] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[[[]]]] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[[]],[],[]] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[[]],[[]]] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[[[]]],[]] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[],[[]],[],[],[]] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[],[[]],[],[[]]] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[],[[]],[[]],[]] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[],[[]],[[[]]]] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[],[[[]]],[],[]] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[],[[[]]],[[]]] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[],[[[[]]]],[]] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[],[[[[[]]]]]] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[]] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[[]]] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[[]],[],[[]],[]] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[[]],[],[[[]]]] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[[]],[[]],[],[]] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[[]],[[]],[[]]] => 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[[]],[[[]]],[]] => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[[]],[[[[]]]]] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[[[]]],[],[],[]] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[[[]]],[],[[]]] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[[[]]],[[]],[]] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[[[]]],[[[]]]] => 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[[]]]],[],[]] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[[[]]]],[[]]] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[[]]]]],[]] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[[[[[]]]]]]] => 6
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[],[],[],[],[],[],[]] => 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[],[],[],[],[],[[]]] => 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[],[],[],[],[[]],[]] => 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[],[],[],[],[[[]]]] => 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[],[],[],[[]],[],[]] => 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[],[],[],[[]],[[]]] => 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[],[],[],[[[]]],[]] => 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[],[[[[]]]]] => 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[],[],[[]],[],[],[]] => 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[],[],[[]],[],[[]]] => 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[],[],[[]],[[]],[]] => 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[],[],[[]],[[[]]]] => 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[],[],[[[]]],[],[]] => 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[],[],[[[]]],[[]]] => 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[],[],[[[[]]]],[]] => 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[],[],[[[[[]]]]]] => 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[],[[]],[],[],[],[]] => 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[],[[]],[],[],[[]]] => 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[],[[]],[],[[]],[]] => 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[],[[]],[],[[[]]]] => 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[],[[]],[[]],[],[]] => 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]],[[]]] => 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[],[[]],[[[]]],[]] => 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[],[[]],[[[[]]]]] => 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[],[[[]]],[],[],[]] => 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[],[[[]]],[],[[]]] => 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[],[[[]]],[[]],[]] => 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[],[[[]]],[[[]]]] => 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[],[[[[]]]],[],[]] => 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[],[[[[]]]],[[]]] => 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[],[[[[[]]]]],[]] => 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[],[[[[[[]]]]]]] => 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[[]],[],[],[],[],[]] => 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[[]],[],[],[],[[]]] => 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[[]],[],[],[[]],[]] => 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[[]],[],[],[[[]]]] => 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[[]],[],[[]],[],[]] => 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[[]],[],[[]],[[]]] => 1
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Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
This is the minimal distance from the root to a leaf.
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.
Map
bounce path
Description
The bounce path determined by an integer composition.
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