Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000120: Dyck paths ⟶ ℤ
Values
[[]] => [1,0] => [1,0] => 0
[[],[]] => [1,0,1,0] => [1,1,0,0] => 0
[[[]]] => [1,1,0,0] => [1,0,1,0] => 1
[[],[],[]] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
[[],[[]]] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 1
[[[]],[]] => [1,1,0,0,1,0] => [1,0,1,0,1,0] => 1
[[[],[]]] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 1
[[[[]]]] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => 2
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 1
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0] => 1
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => 1
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0] => 2
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 1
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => 3
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 2
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 2
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 2
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => 2
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 2
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 3
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 1
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => 2
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 2
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 2
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 3
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 1
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => 2
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => 1
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 2
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 2
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => 3
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 2
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 2
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 2
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 2
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 3
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 3
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => 3
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 3
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => 3
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 4
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 1
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 1
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 1
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 1
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 3
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 1
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 2
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 2
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 3
[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 1
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 2
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 3
[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 1
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 2
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 2
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 2
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 3
[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 1
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 2
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 2
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 2
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 3
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 2
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 2
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 3
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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.
Map
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
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