- FindStat

Identifier

Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤ (values match St000166The depth minus 1 of an ordered tree.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,0,0,0,0] => [[[[[[]]]]]] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 631 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The depth of an ordered 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.