Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1]] => {{1}} => 0
[1,0,1,0,1,0] => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 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}} => 0
[1,1,0,1,0,0] => [1] => [[1]] => {{1}} => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 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] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 0
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 0
[1,1,0,1,0,0,1,0] => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 0
[1,1,0,1,0,1,0,0] => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 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}} => 0
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => {{1,2}} => 0
[1,1,1,0,1,0,0,0] => [1] => [[1]] => {{1}} => 0
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => {{1,2},{3,4},{5,6},{7}} => 0
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => {{1,2,3},{4,5},{6},{7}} => 0
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 0
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => {{1,2,3,4},{5},{6},{7}} => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 0
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 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] => [[1,2,3],[4,5],[6,7]] => {{1,2,3},{4,5},{6,7}} => 0
[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}} => 0
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => {{1,2,3},{4,5,6},{7}} => 0
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => {{1,2,3,4},{5,6},{7}} => 0
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 0
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 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] => [[1,2,3,4],[5,6,7]] => {{1,2,3,4},{5,6,7}} => 0
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 0
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,2,3,4],[5]] => {{1,2,3,4},{5}} => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 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}} => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => {{1,2,3}} => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => {{1,2}} => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1]] => {{1}} => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => {{1,2},{3,4},{5},{6},{7}} => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => {{1,2,3},{4},{5},{6},{7}} => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 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
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => {{1,2},{3,4},{5,6},{7,8}} => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => {{1,2},{3,4},{5,6},{7}} => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => {{1,2,3},{4,5},{6},{7}} => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => {{1,2,3,4},{5},{6},{7}} => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 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] => [[1,2,3],[4,5],[6,7]] => {{1,2,3},{4,5},{6,7}} => 0
[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}} => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [[1,2,3],[4,5,6],[7]] => {{1,2,3},{4,5,6},{7}} => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[1,2,3,4],[5,6],[7]] => {{1,2,3,4},{5,6},{7}} => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[1,2,3,4,5],[6],[7]] => {{1,2,3,4,5},{6},{7}} => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 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] => [[1,2,3,4],[5,6,7]] => {{1,2,3,4},{5,6,7}} => 0
[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}} => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[1,2,3,4,5],[6,7]] => {{1,2,3,4,5},{6,7}} => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[1,2,3,4,5],[6]] => {{1,2,3,4,5},{6}} => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,2,3,4],[5]] => {{1,2,3,4},{5}} => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 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}} => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => {{1,2,3,4}} => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => {{1,2,3}} => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => {{1,2}} => 0
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1]] => {{1}} => 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => {{1,2},{3},{4},{5},{6},{7}} => 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
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => {{1,2},{3,4},{5,6},{7,8},{9,10}} => 0
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => {{1,2},{3,4},{5},{6},{7}} => 0
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => {{1,2,3},{4},{5},{6},{7}} => 0
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 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
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => {{1,2},{3,4},{5,6},{7,8}} => 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => {{1,2},{3,4},{5,6},{7}} => 0
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => {{1,2,3},{4,5},{6},{7}} => 0
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 0
[1,1,1,0,1,1,1,0,0,0,1,0,0,0] => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => {{1,2,3,4},{5},{6},{7}} => 0
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 0
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 0
>>> Load all 283 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of crossings of a set partition.
This is given by the number of i<i′<j<j′ such that i,j are two consecutive entries on one block, and i′,j′ are consecutive entries in another block.
This is given by the number of i<i′<j<j′ such that i,j are two consecutive entries on one block, and i′,j′ are consecutive entries in another block.
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).
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers 1 through n row by row.
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!