Identifier
-
Mp00083:
Standard tableaux
—shape⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤ
Values
[[1]] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 1
[[1,2]] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[[1],[2]] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 1
[[1,2,3]] => [3] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 1
[[1,3],[2]] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[[1,2],[3]] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[[1],[2],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 1
[[1,3,4],[2]] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[[1,2,4],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[[1,2,3],[4]] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[[1,3],[2,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 1
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 1
[[1,4],[2],[3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[[1,3],[2],[4]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[[1,2],[3],[4]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[[1,3,5],[2,4]] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[[1,2,5],[3,4]] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[[1,3,4],[2,5]] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[[1,2,4],[3,5]] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[[1,2,3],[4,5]] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[[1,4,5],[2],[3]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[[1,3,5],[2],[4]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[[1,2,5],[3],[4]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[[1,3,4],[2],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[[1,2,4],[3],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[[1,2,3],[4],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[[1,4],[2,5],[3]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[[1,3],[2,5],[4]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[[1,2],[3,5],[4]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[[1,3],[2,4],[5]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[[1,2],[3,4],[5]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[[1,4,6],[2,5],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,3,6],[2,5],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,6],[3,5],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,3,6],[2,4],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,6],[3,4],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,4,5],[2,6],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,3,5],[2,6],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,5],[3,6],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,3,4],[2,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,4],[3,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,3],[4,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,3,5],[2,4],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,5],[3,4],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,3,4],[2,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,4],[3,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
[[1,2,3],[4,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
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Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Map
shape
Description
Sends a tableau to its shape.
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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