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
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.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition λ associated to a Dyck path is defined to be the complementary partition inside the staircase partition (n−1,…,2,1) when cutting out D considered as a path from (0,0) to (n,n).
In other words, λi 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,…,2,1).
The partition λ associated to a Dyck path is defined to be the complementary partition inside the staircase partition (n−1,…,2,1) when cutting out D considered as a path from (0,0) to (n,n).
In other words, λi 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,…,2,1).
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