Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: Ordered trees ⟶ ℤ
Values
[1] => [1,0,1,0] => [[],[]] => 2
[2] => [1,1,0,0,1,0] => [[[]],[]] => 2
[1,1] => [1,0,1,1,0,0] => [[],[[]]] => 2
[3] => [1,1,1,0,0,0,1,0] => [[[[]]],[]] => 2
[2,1] => [1,0,1,0,1,0] => [[],[],[]] => 3
[1,1,1] => [1,0,1,1,1,0,0,0] => [[],[[[]]]] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [[[[[]]]],[]] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [[[],[]],[]] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [[[]],[[]]] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [[],[[],[]]] => 3
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [[],[[[[]]]]] => 2
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[[[[[]]]]],[]] => 2
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [[[[],[]]],[]] => 3
[3,2] => [1,1,0,0,1,0,1,0] => [[[]],[],[]] => 3
[3,1,1] => [1,0,1,1,0,0,1,0] => [[],[[]],[]] => 3
[2,2,1] => [1,0,1,0,1,1,0,0] => [[],[],[[]]] => 3
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => 3
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[],[[[[[]]]]]] => 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [[[[[[[]]]]]],[]] => 2
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [[[[[],[]]]],[]] => 3
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [[[[]],[]],[]] => 3
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [[[],[[]]],[]] => 3
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [[[[]]],[[]]] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [[],[],[],[]] => 4
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => 3
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [[[]],[[[]]]] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => 3
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [[],[[[[],[]]]]] => 3
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[],[[[[[[]]]]]]] => 2
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[[[[[[[]]]]]]],[]] => 2
[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [[[[[[],[]]]]],[]] => 3
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [[[[[]],[]]],[]] => 3
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [[[[],[[]]]],[]] => 3
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [[[[]]],[],[]] => 3
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => 4
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => 3
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [[[]],[[],[]]] => 3
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => 4
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [[],[[[[]],[]]]] => 3
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [[],[],[[[]]]] => 3
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => 3
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [[],[[[[[],[]]]]]] => 3
[1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[],[[[[[[[]]]]]]]] => 2
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[]]]]]]]],[]] => 2
[7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [[[[[[[],[]]]]]],[]] => 3
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [[[[[[]],[]]]],[]] => 3
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [[[[[],[[]]]]],[]] => 3
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [[[[[]]],[]],[]] => 3
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [[[[],[],[]]],[]] => 4
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [[[],[[[]]]],[]] => 3
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[[[[]]]],[[]]] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => 4
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [[[]],[[]],[]] => 3
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [[],[[],[]],[]] => 4
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [[],[[[[]]],[]]] => 3
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [[[]],[],[[]]] => 3
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [[],[[]],[[]]] => 3
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [[],[],[[],[]]] => 4
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [[],[[[],[],[]]]] => 4
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [[],[[[[[]],[]]]]] => 3
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[[]],[[[[]]]]] => 2
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [[],[[],[[[]]]]] => 3
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [[],[[[[],[[]]]]]] => 3
[2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [[],[[[[[[],[]]]]]]] => 3
[1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[],[[[[[[[[]]]]]]]]] => 2
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[[]]]]]]]]],[]] => 2
[8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [[[[[[[[],[]]]]]]],[]] => 3
[7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [[[[[[[]],[]]]]],[]] => 3
[7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [[[[[[],[[]]]]]],[]] => 3
[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[[[[[]]],[]]],[]] => 3
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [[[[[],[],[]]]],[]] => 4
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [[[[],[[[]]]]],[]] => 3
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[[[[]]]],[],[]] => 3
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [[[[],[]],[]],[]] => 4
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [[[[]],[[]]],[]] => 3
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [[[],[[],[]]],[]] => 4
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[],[[[[]]]],[]] => 3
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [[[[],[]]],[[]]] => 3
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [[[]],[],[],[]] => 4
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[],[[]],[],[]] => 4
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[],[],[[]],[]] => 4
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [[],[[[],[]],[]]] => 4
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [[],[[[[[]]],[]]]] => 3
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[[[]]],[[[]]]] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [[],[],[],[[]]] => 4
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => 3
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [[[]],[[[],[]]]] => 3
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [[],[[],[[],[]]]] => 4
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [[],[[[[],[],[]]]]] => 4
[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [[],[[[[[[]],[]]]]]] => 3
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[],[],[[[[]]]]] => 3
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [[],[[[],[[[]]]]]] => 3
[2,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [[],[[[[[],[[]]]]]]] => 3
[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [[],[[[[[[[],[]]]]]]]] => 3
[1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [[],[[[[[[[[[]]]]]]]]]] => 2
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[[[]]]]]]]]]],[]] => 2
[9,1] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [[[[[[[[[],[]]]]]]]],[]] => 3
[8,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [[[[[[[[]],[]]]]]],[]] => 3
[8,1,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [[[[[[[],[[]]]]]]],[]] => 3
[7,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [[[[[[[]]],[]]]],[]] => 3
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Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
This is the number of nodes which do not have any children.
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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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