Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000502: Set partitions ⟶ ℤ
Values
[1,0,1,0,1,0] => [2,1] => [[1,3],[2]] => {{1,3},{2}} => 0
[1,0,1,1,0,0] => [1,1] => [[1],[2]] => {{1},{2}} => 0
[1,1,0,0,1,0] => [2] => [[1,2]] => {{1,2}} => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => [[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,4],[2],[3]] => {{1,4},{2},{3}} => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 0
[1,1,0,0,1,0,1,0] => [3,2] => [[1,2,5],[3,4]] => {{1,2,5},{3,4}} => 2
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 2
[1,1,0,1,0,0,1,0] => [3,1] => [[1,3,4],[2]] => {{1,3,4},{2}} => 1
[1,1,0,1,0,1,0,0] => [2,1] => [[1,3],[2]] => {{1,3},{2}} => 0
[1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => {{1},{2}} => 0
[1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => {{1,2,3}} => 2
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => {{1,2}} => 1
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => {{1,3},{2,5},{4,7},{6}} => 0
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => {{1,4,7},{2,6},{3},{5}} => 0
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => {{1,4},{2,6},{3},{5}} => 0
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => {{1,5,6,7},{2},{3},{4}} => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => 1
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 0
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => {{1,2,7},{3,4},{5,6}} => 3
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 3
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => {{1,3,4},{2,6,7},{5}} => 2
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => {{1,3,6,7},{2,5},{4}} => 1
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => 0
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => {{1,4,5,6},{2},{3}} => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => {{1,4},{2},{3}} => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 0
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [[1,2,3,7],[4,5,6]] => {{1,2,3,7},{4,5,6}} => 4
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 4
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,5,6],[3,4]] => {{1,2,5,6},{3,4}} => 3
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,5],[3,4]] => {{1,2,5},{3,4}} => 2
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,3,4,5],[2]] => {{1,3,4,5},{2}} => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,3,4],[2]] => {{1,3,4},{2}} => 1
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,3],[2]] => {{1,3},{2}} => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => {{1},{2}} => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => {{1,2,3,4}} => 3
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => {{1,2,3}} => 2
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => {{1,2}} => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => {{1,5},{2,7},{3},{4},{6}} => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => {{1,6,7},{2},{3},{4},{5}} => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => {{1,6},{2},{3},{4},{5}} => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => {{1,3},{2,5},{4,7},{6}} => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => {{1,4,7},{2,6},{3},{5}} => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => {{1,4},{2,6},{3},{5}} => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => {{1,5,6,7},{2},{3},{4}} => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => {{1,2,7},{3,4},{5,6}} => 3
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 3
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => {{1,3,4},{2,6,7},{5}} => 2
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => {{1,3,6,7},{2,5},{4}} => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => {{1,4,5,6,7},{2},{3}} => 3
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => {{1,4,5,6},{2},{3}} => 2
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => {{1,4},{2},{3}} => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [[1,2,3,7],[4,5,6]] => {{1,2,3,7},{4,5,6}} => 4
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 4
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[1,2,5,6,7],[3,4]] => {{1,2,5,6,7},{3,4}} => 4
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,5,6],[3,4]] => {{1,2,5,6},{3,4}} => 3
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,5],[3,4]] => {{1,2,5},{3,4}} => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[1,3,4,5,6],[2]] => {{1,3,4,5,6},{2}} => 3
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,3,4,5],[2]] => {{1,3,4,5},{2}} => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,3,4],[2]] => {{1,3,4},{2}} => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,3],[2]] => {{1,3},{2}} => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => {{1},{2}} => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[1,2,3,4,5]] => {{1,2,3,4,5}} => 4
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => {{1,2,3,4}} => 3
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => {{1,2,3}} => 2
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => {{1,2}} => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => {{1,7},{2},{3},{4},{5},{6}} => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => {{1},{2},{3},{4},{5},{6}} => 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => {{1,5},{2,7},{3},{4},{6}} => 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => {{1,6,7},{2},{3},{4},{5}} => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => {{1,6},{2},{3},{4},{5}} => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => {{1,3},{2,5},{4,7},{6}} => 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => {{1,4,7},{2,6},{3},{5}} => 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => {{1,4},{2,6},{3},{5}} => 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => {{1,5,6,7},{2},{3},{4}} => 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 0
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => {{1,2,7},{3,4},{5,6}} => 3
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => {{1,3,4},{2,6,7},{5}} => 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => {{1,3,6,7},{2,5},{4}} => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => 0
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,1,1] => [[1,4,5,6,7,8],[2],[3]] => {{1,4,5,6,7,8},{2},{3}} => 4
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Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
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