Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000035: Permutations ⟶ ℤ
Values
[] => [] => [] => 0
[[]] => [1,0] => [1] => 0
[[],[]] => [1,0,1,0] => [1,2] => 0
[[[]]] => [1,1,0,0] => [2,1] => 1
[[],[],[]] => [1,0,1,0,1,0] => [1,2,3] => 0
[[],[[]]] => [1,0,1,1,0,0] => [1,3,2] => 1
[[[]],[]] => [1,1,0,0,1,0] => [2,1,3] => 1
[[[],[]]] => [1,1,0,1,0,0] => [2,3,1] => 1
[[[[]]]] => [1,1,1,0,0,0] => [3,2,1] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 1
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 1
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 1
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 2
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 1
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 1
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 1
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => 1
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 1
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 1
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 1
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 1
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 2
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 2
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 1
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 2
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 1
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 2
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 1
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 1
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 2
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 2
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 1
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 1
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 1
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 2
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 2
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 1
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 2
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 2
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 2
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 2
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 1
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 1
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 1
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 1
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 1
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 2
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 1
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 1
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 1
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 1
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 2
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => 2
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 1
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 1
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 2
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 2
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 2
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 2
[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 1
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 1
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 2
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 2
[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 1
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 1
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 2
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => 2
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 1
[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 1
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 1
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 2
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => 2
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 1
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 2
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 2
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => 2
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Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with St000366The number of double descents of a permutation. is studied in [3], where left outer peaks are called exterior peaks.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with St000366The number of double descents of a permutation. is studied in [3], where left outer peaks are called exterior peaks.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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.
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