Identifier
-
Mp00103:
Dyck paths
—peeling map⟶
Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤ
Values
[1,0] => [1,0] => [[1],[2]] => 0
[1,0,1,0] => [1,0,1,0] => [[1,3],[2,4]] => 0
[1,1,0,0] => [1,0,1,0] => [[1,3],[2,4]] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[1,0,1,1,0,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[1,1,0,0,1,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 0
[1,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => 1
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => 1
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => 3
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => 1
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Description
The Shynar inversion number of a standard tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Map
peeling map
Description
Send a Dyck path to its peeled Dyck path.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
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