- FindStat

Identifier

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000974: Ordered trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 1 + q^{2}$

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

$F_{4} = 9 + 3\ q + q^{2} + q^{4}$

$F_{5} = 28 + 9\ q + 3\ q^{2} + q^{3} + q^{5}$

$F_{6} = 90 + 28\ q + 9\ q^{2} + 3\ q^{3} + q^{4} + q^{6}$

$F_{7} = 297 + 90\ q + 28\ q^{2} + 9\ q^{3} + 3\ q^{4} + q^{5} + q^{7}$

Description

The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.

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.