Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001174: Permutations ⟶ ℤ
Values
[[],[]] => [1,0,1,0] => [1,2] => 0
[[[]]] => [1,1,0,0] => [2,1] => 0
[[],[],[]] => [1,0,1,0,1,0] => [1,2,3] => 0
[[],[[]]] => [1,0,1,1,0,0] => [1,3,2] => 0
[[[]],[]] => [1,1,0,0,1,0] => [2,1,3] => 0
[[[],[]]] => [1,1,0,1,0,0] => [2,3,1] => 1
[[[[]]]] => [1,1,1,0,0,0] => [3,2,1] => 0
[[],[],[],[]] => [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] => 0
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[[],[[],[]]] => [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] => 0
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 0
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 0
[[[],[]],[]] => [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] => 0
[[[],[],[]]] => [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] => 1
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => 1
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[[],[],[],[],[]] => [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] => 0
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
[[],[],[[],[]]] => [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] => 0
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0
[[],[[],[]],[]] => [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] => 0
[[],[[],[],[]]] => [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] => 1
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 1
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 0
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 0
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 0
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 1
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 0
[[[],[]],[],[]] => [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] => 0
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 1
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 0
[[[],[],[]],[]] => [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] => 1
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 1
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
[[[],[],[],[]]] => [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] => 1
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 1
[[[],[[[]]]]] => [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] => 1
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 1
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 1
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 1
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 1
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 1
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 1
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[[],[],[],[],[],[]] => [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] => 0
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
[[],[],[],[[],[]]] => [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] => 0
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 0
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 0
[[],[],[[],[]],[]] => [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] => 0
[[],[],[[],[],[]]] => [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] => 1
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => 1
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 0
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 0
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 1
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 0
[[],[[],[]],[],[]] => [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] => 0
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 1
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 0
[[],[[],[],[]],[]] => [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] => 1
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => 1
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 0
[[],[[],[],[],[]]] => [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] => 1
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => 1
[[],[[],[[[]]]]] => [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] => 1
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 1
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => 1
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => 1
[[],[[[],[],[]]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => 1
>>> Load all 195 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 Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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.
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!