- FindStat

Identifier

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000167: 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] => [[],[[],[]]] => 3

[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] => [[[],[]],[]] => 3

[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] => [[],[],[[],[]]] => 4

[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] => [[],[[],[]],[]] => 4

[1,0,1,1,0,1,0,1,0,0] => [[],[[],[],[]]] => 4

[1,0,1,1,0,1,1,0,0,0] => [[],[[],[[]]]] => 3

[1,0,1,1,1,0,0,0,1,0] => [[],[[[]]],[]] => 3

[1,0,1,1,1,0,0,1,0,0] => [[],[[[]],[]]] => 3

[1,0,1,1,1,0,1,0,0,0] => [[],[[[],[]]]] => 3

[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] => [[[]],[[],[]]] => 3

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

[1,1,0,1,0,0,1,0,1,0] => [[[],[]],[],[]] => 4

[1,1,0,1,0,0,1,1,0,0] => [[[],[]],[[]]] => 3

[1,1,0,1,0,1,0,0,1,0] => [[[],[],[]],[]] => 4

[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] => [[[],[[]]],[]] => 3

[1,1,0,1,1,0,0,1,0,0] => [[[],[[]],[]]] => 3

[1,1,0,1,1,0,1,0,0,0] => [[[],[[],[]]]] => 3

[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] => [[[[]],[]],[]] => 3

[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] => [[[[],[]]],[]] => 3

[1,1,1,0,1,0,0,1,0,0] => [[[[],[]],[]]] => 3

[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] => [[],[],[],[[],[]]] => 5

[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] => [[],[],[[],[]],[]] => 5

[1,0,1,0,1,1,0,1,0,1,0,0] => [[],[],[[],[],[]]] => 5

[1,0,1,0,1,1,0,1,1,0,0,0] => [[],[],[[],[[]]]] => 4

[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] => [[],[],[[[]],[]]] => 4

[1,0,1,0,1,1,1,0,1,0,0,0] => [[],[],[[[],[]]]] => 4

[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] => [[],[[]],[[],[]]] => 4

[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] => [[],[[],[]],[],[]] => 5

[1,0,1,1,0,1,0,0,1,1,0,0] => [[],[[],[]],[[]]] => 4

[1,0,1,1,0,1,0,1,0,0,1,0] => [[],[[],[],[]],[]] => 5

[1,0,1,1,0,1,0,1,0,1,0,0] => [[],[[],[],[],[]]] => 5

[1,0,1,1,0,1,0,1,1,0,0,0] => [[],[[],[],[[]]]] => 4

[1,0,1,1,0,1,1,0,0,0,1,0] => [[],[[],[[]]],[]] => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => [[],[[],[[]],[]]] => 4

[1,0,1,1,0,1,1,0,1,0,0,0] => [[],[[],[[],[]]]] => 4

[1,0,1,1,0,1,1,1,0,0,0,0] => [[],[[],[[[]]]]] => 3

[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] => [[],[[[]],[]],[]] => 4

[1,0,1,1,1,0,0,1,0,1,0,0] => [[],[[[]],[],[]]] => 4

[1,0,1,1,1,0,0,1,1,0,0,0] => [[],[[[]],[[]]]] => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => [[],[[[],[]]],[]] => 4

[1,0,1,1,1,0,1,0,0,1,0,0] => [[],[[[],[]],[]]] => 4

[1,0,1,1,1,0,1,0,1,0,0,0] => [[],[[[],[],[]]]] => 4

[1,0,1,1,1,0,1,1,0,0,0,0] => [[],[[[],[[]]]]] => 3

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 3\ q^{2} + q^{3}$

$F_{4} = q + 6\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = q + 10\ q^{2} + 20\ q^{3} + 10\ q^{4} + q^{5}$

$F_{6} = q + 15\ q^{2} + 50\ q^{3} + 50\ q^{4} + 15\ q^{5} + q^{6}$

Description

The number of leaves of an ordered tree.
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.