Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
St000120: Dyck paths ⟶ ℤ
Values
[[]] => [1,0] => 0
[[],[]] => [1,0,1,0] => 1
[[[]]] => [1,1,0,0] => 0
[[],[],[]] => [1,0,1,0,1,0] => 1
[[],[[]]] => [1,0,1,1,0,0] => 1
[[[]],[]] => [1,1,0,0,1,0] => 2
[[[],[]]] => [1,1,0,1,0,0] => 1
[[[[]]]] => [1,1,1,0,0,0] => 0
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => 2
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => 2
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => 1
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => 1
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => 2
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => 2
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => 3
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => 1
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => 1
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => 1
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => 0
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 2
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => 2
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => 2
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => 2
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => 2
[[],[[]],[],[]] => [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] => 2
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => 1
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => 2
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => 2
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => 1
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => 1
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => 2
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => 2
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => 2
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => 2
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => 3
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => 3
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => 3
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => 3
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => 3
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => 3
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => 3
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => 4
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => 2
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => 2
[[[],[[]],[]]] => [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
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => 2
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => 3
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => 1
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => 1
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => 1
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => 0
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 3
[[],[],[],[],[[]]] => [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] => 3
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 2
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => 2
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[[],[[]],[],[],[]] => [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] => 3
[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 3
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => 3
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 4
[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 2
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => 2
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => 3
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => 2
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1
[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => 2
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => 2
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => 2
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => 2
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 2
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => 3
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 3
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => 2
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 1
>>> Load all 196 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
Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b<n then the tunnel is called left.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b<n then the tunnel is called left.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
This sends the maximal height of the Dyck path to the depth of the tree.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!