Identifier
-
Mp00033:
Dyck paths
—to two-row standard tableau⟶
Standard tableaux
St000738: Standard tableaux ⟶ ℤ
Values
[1,0] => [[1],[2]] => 2
[1,0,1,0] => [[1,3],[2,4]] => 2
[1,1,0,0] => [[1,2],[3,4]] => 3
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 3
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 3
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 4
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 3
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 3
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 3
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 3
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 4
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 4
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 4
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 5
[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 2
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => 2
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => 2
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => 2
[1,0,1,0,1,1,1,0,0,0] => [[1,3,5,6,7],[2,4,8,9,10]] => 2
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 2
[1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => 2
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => 2
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => 2
[1,0,1,1,0,1,1,0,0,0] => [[1,3,4,6,7],[2,5,8,9,10]] => 2
[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => 2
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => 2
[1,0,1,1,1,0,1,0,0,0] => [[1,3,4,5,7],[2,6,8,9,10]] => 2
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => 2
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => 3
[1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => 3
[1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => 3
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => 3
[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => 3
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => 3
[1,1,0,1,0,0,1,1,0,0] => [[1,2,4,7,8],[3,5,6,9,10]] => 3
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => 3
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => 3
[1,1,0,1,0,1,1,0,0,0] => [[1,2,4,6,7],[3,5,8,9,10]] => 3
[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => 3
[1,1,0,1,1,0,0,1,0,0] => [[1,2,4,5,8],[3,6,7,9,10]] => 3
[1,1,0,1,1,0,1,0,0,0] => [[1,2,4,5,7],[3,6,8,9,10]] => 3
[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => 3
[1,1,1,0,0,0,1,0,1,0] => [[1,2,3,7,9],[4,5,6,8,10]] => 4
[1,1,1,0,0,0,1,1,0,0] => [[1,2,3,7,8],[4,5,6,9,10]] => 4
[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => 4
[1,1,1,0,0,1,0,1,0,0] => [[1,2,3,6,8],[4,5,7,9,10]] => 4
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => 4
[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => 4
[1,1,1,0,1,0,0,1,0,0] => [[1,2,3,5,8],[4,6,7,9,10]] => 4
[1,1,1,0,1,0,1,0,0,0] => [[1,2,3,5,7],[4,6,8,9,10]] => 4
[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => 4
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => 5
[1,1,1,1,0,0,0,1,0,0] => [[1,2,3,4,8],[5,6,7,9,10]] => 5
[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => 5
[1,1,1,1,0,1,0,0,0,0] => [[1,2,3,4,6],[5,7,8,9,10]] => 5
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [[1,3,5,7,8,9],[2,4,6,10,11,12]] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [[1,3,5,6,9,10],[2,4,7,8,11,12]] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [[1,3,5,6,8,11],[2,4,7,9,10,12]] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [[1,3,5,6,8,9],[2,4,7,10,11,12]] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,6,7,11],[2,4,8,9,10,12]] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [[1,3,5,6,7,10],[2,4,8,9,11,12]] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [[1,3,5,6,7,9],[2,4,8,10,11,12]] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [[1,3,5,6,7,8],[2,4,9,10,11,12]] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [[1,3,4,7,9,10],[2,5,6,8,11,12]] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,4,7,8,11],[2,5,6,9,10,12]] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [[1,3,4,7,8,10],[2,5,6,9,11,12]] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [[1,3,4,7,8,9],[2,5,6,10,11,12]] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [[1,3,4,6,9,11],[2,5,7,8,10,12]] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [[1,3,4,6,9,10],[2,5,7,8,11,12]] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [[1,3,4,6,8,11],[2,5,7,9,10,12]] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [[1,3,4,6,8,9],[2,5,7,10,11,12]] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [[1,3,4,6,7,11],[2,5,8,9,10,12]] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [[1,3,4,6,7,10],[2,5,8,9,11,12]] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [[1,3,4,6,7,9],[2,5,8,10,11,12]] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [[1,3,4,6,7,8],[2,5,9,10,11,12]] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [[1,3,4,5,9,11],[2,6,7,8,10,12]] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [[1,3,4,5,9,10],[2,6,7,8,11,12]] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [[1,3,4,5,8,11],[2,6,7,9,10,12]] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [[1,3,4,5,8,10],[2,6,7,9,11,12]] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [[1,3,4,5,8,9],[2,6,7,10,11,12]] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [[1,3,4,5,7,11],[2,6,8,9,10,12]] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [[1,3,4,5,7,10],[2,6,8,9,11,12]] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [[1,3,4,5,7,9],[2,6,8,10,11,12]] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [[1,3,4,5,7,8],[2,6,9,10,11,12]] => 2
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Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see St000734The last entry in the first row of a standard tableau..
For the last entry in the first row, see St000734The last entry in the first row of a standard tableau..
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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