Identifier
Values
([2],3) => [2] => [[1,2]] => {{1,2}} => 1
([1,1],3) => [1,1] => [[1],[2]] => {{1},{2}} => 1
([3,1],3) => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 3
([2,1,1],3) => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 3
([4,2],3) => [4,2] => [[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => 5
([3,1,1],3) => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 4
([2,2,1,1],3) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 5
([2],4) => [2] => [[1,2]] => {{1,2}} => 1
([1,1],4) => [1,1] => [[1],[2]] => {{1},{2}} => 1
([3],4) => [3] => [[1,2,3]] => {{1,2,3}} => 2
([2,1],4) => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 2
([1,1,1],4) => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 2
([4,1],4) => [4,1] => [[1,2,3,4],[5]] => {{1,2,3,4},{5}} => 4
([2,2],4) => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 3
([3,1,1],4) => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 4
([2,1,1,1],4) => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 4
([5,2],4) => [5,2] => [[1,2,3,4,5],[6,7]] => {{1,2,3,4,5},{6,7}} => 6
([4,1,1],4) => [4,1,1] => [[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => 5
([3,2,1],4) => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 5
([3,1,1,1],4) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 5
([2,2,1,1,1],4) => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => {{1,2},{3,4},{5},{6},{7}} => 6
([4,1,1,1],4) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => {{1,2,3,4},{5},{6},{7}} => 6
([2],5) => [2] => [[1,2]] => {{1,2}} => 1
([1,1],5) => [1,1] => [[1],[2]] => {{1},{2}} => 1
([3],5) => [3] => [[1,2,3]] => {{1,2,3}} => 2
([2,1],5) => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 2
([1,1,1],5) => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 2
([4],5) => [4] => [[1,2,3,4]] => {{1,2,3,4}} => 3
([3,1],5) => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 3
([2,2],5) => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 3
([2,1,1],5) => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 3
([1,1,1,1],5) => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 3
([5,1],5) => [5,1] => [[1,2,3,4,5],[6]] => {{1,2,3,4,5},{6}} => 5
([3,2],5) => [3,2] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 4
([4,1,1],5) => [4,1,1] => [[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => 5
([2,2,1],5) => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 4
([3,1,1,1],5) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 5
([2,1,1,1,1],5) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 5
([5,1,1],5) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => {{1,2,3,4,5},{6},{7}} => 6
([3,3],5) => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 5
([4,2,1],5) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => {{1,2,3,4},{5,6},{7}} => 6
([4,1,1,1],5) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => {{1,2,3,4},{5},{6},{7}} => 6
([2,2,2],5) => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 5
([3,2,1,1],5) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => {{1,2,3},{4,5},{6},{7}} => 6
([3,1,1,1,1],5) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => {{1,2,3},{4},{5},{6},{7}} => 6
([2],6) => [2] => [[1,2]] => {{1,2}} => 1
([1,1],6) => [1,1] => [[1],[2]] => {{1},{2}} => 1
([3],6) => [3] => [[1,2,3]] => {{1,2,3}} => 2
([2,1],6) => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 2
([1,1,1],6) => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 2
([4],6) => [4] => [[1,2,3,4]] => {{1,2,3,4}} => 3
([3,1],6) => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 3
([2,2],6) => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 3
([2,1,1],6) => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 3
([1,1,1,1],6) => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 3
([5],6) => [5] => [[1,2,3,4,5]] => {{1,2,3,4,5}} => 4
([4,1],6) => [4,1] => [[1,2,3,4],[5]] => {{1,2,3,4},{5}} => 4
([3,2],6) => [3,2] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 4
([3,1,1],6) => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 4
([2,2,1],6) => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 4
([2,1,1,1],6) => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 4
([1,1,1,1,1],6) => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 4
([6,1],6) => [6,1] => [[1,2,3,4,5,6],[7]] => {{1,2,3,4,5,6},{7}} => 6
([4,2],6) => [4,2] => [[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => 5
([5,1,1],6) => [5,1,1] => [[1,2,3,4,5],[6],[7]] => {{1,2,3,4,5},{6},{7}} => 6
([3,3],6) => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 5
([3,2,1],6) => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 5
([4,1,1,1],6) => [4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => {{1,2,3,4},{5},{6},{7}} => 6
([2,2,2],6) => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 5
([2,2,1,1],6) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 5
([3,1,1,1,1],6) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => {{1,2,3},{4},{5},{6},{7}} => 6
([2,1,1,1,1,1],6) => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => {{1,2},{3},{4},{5},{6},{7}} => 6
([4,3],6) => [4,3] => [[1,2,3,4],[5,6,7]] => {{1,2,3,4},{5,6,7}} => 6
([3,3,1],6) => [3,3,1] => [[1,2,3],[4,5,6],[7]] => {{1,2,3},{4,5,6},{7}} => 6
([3,2,2],6) => [3,2,2] => [[1,2,3],[4,5],[6,7]] => {{1,2,3},{4,5},{6,7}} => 6
([2,2,2,1],6) => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => {{1,2},{3,4},{5,6},{7}} => 6
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Description
The number of ascent tops in the flattened set partition such that all smaller elements appear before.
Let $P$ be a set partition. The flattened set partition is the permutation obtained by sorting the set of blocks of $P$ according to their minimal element and the elements in each block in increasing order.
Given a set partition $P$, this statistic is the binary logarithm of the number of set partitions that flatten to the same permutation as $P$.
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
to partition
Description
Considers a core as a partition.
This embedding is graded and injective but not surjective on $k$-cores for a given parameter $k$, while it is surjective and neither graded nor injective on the collection of all cores.