- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations

Images

[] => [] => []

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

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

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

[[],[],[]] => [1,0,1,0,1,0] => [3,2,1]

[[],[[]]] => [1,0,1,1,0,0] => [2,3,1]

[[[]],[]] => [1,1,0,0,1,0] => [3,1,2]

[[[],[]]] => [1,1,0,1,0,0] => [2,1,3]

[[[[]]]] => [1,1,1,0,0,0] => [1,2,3]

[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [4,3,2,1]

[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [3,4,2,1]

[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [4,2,3,1]

[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [3,2,4,1]

[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [2,3,4,1]

[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [4,3,1,2]

[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [3,4,1,2]

[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [4,2,1,3]

[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [4,1,2,3]

[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [3,2,1,4]

[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [2,3,1,4]

[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [3,1,2,4]

[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [2,1,3,4]

[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [1,2,3,4]

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1]

[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1]

[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1]

[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1]

[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1]

[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1]

[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1]

[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1]

[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1]

[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1]

[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1]

[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1]

[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1]

[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1]

[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2]

[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2]

[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2]

[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2]

[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2]

[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3]

[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3]

[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3]

[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3]

[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4]

[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4]

[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4]

[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4]

[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4]

[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5]

[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5]

[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5]

[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5]

[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5]

[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5]

[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5]

[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5]

[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5]

[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5]

[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5]

[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5]

[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5]

[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5]

[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1]

[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1]

[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1]

[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1]

[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1]

[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1]

[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1]

[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1]

[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1]

[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1]

[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1]

[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1]

[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1]

[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1]

[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1]

[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1]

[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1]

[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1]

[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1]

[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1]

[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1]

[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1]

[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1]

[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1]

[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1]

[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1]

[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1]

[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1]

[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1]

[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1]

[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1]

[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1]

[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1]

[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1]

[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1]

[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1]

>>> Load all 2150 entries. <<<

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.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].