Identifier
-
Mp00021:
Cores
—to bounded partition⟶
Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St001641: Set partitions ⟶ ℤ
Values
([2],3) => [2] => [[1,2]] => {{1,2}} => 1
([1,1],3) => [1,1] => [[1],[2]] => {{1},{2}} => 1
([3,1],3) => [2,1] => [[1,2],[3]] => {{1,2},{3}} => 2
([2,1,1],3) => [1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => 2
([4,2],3) => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 3
([3,1,1],3) => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 3
([2,2,1,1],3) => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 3
([5,3,1],3) => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 4
([4,2,1,1],3) => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 4
([3,2,2,1,1],3) => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 4
([6,4,2],3) => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 5
([5,3,1,1],3) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 5
([4,2,2,1,1],3) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 5
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => [[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) => [3,1] => [[1,2,3],[4]] => {{1,2,3},{4}} => 3
([2,2],4) => [2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => 3
([3,1,1],4) => [2,1,1] => [[1,2],[3],[4]] => {{1,2},{3},{4}} => 3
([2,1,1,1],4) => [1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => 3
([5,2],4) => [3,2] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 4
([4,1,1],4) => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 4
([3,2,1],4) => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 4
([3,1,1,1],4) => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 4
([2,2,1,1,1],4) => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 4
([6,3],4) => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 5
([5,2,1],4) => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 5
([4,1,1,1],4) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 5
([4,2,2],4) => [2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => 5
([3,3,1,1],4) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 5
([3,2,1,1,1],4) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 5
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => {{1},{2},{3},{4},{5},{6}} => 5
([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) => [4,1] => [[1,2,3,4],[5]] => {{1,2,3,4},{5}} => 4
([3,2],5) => [3,2] => [[1,2,3],[4,5]] => {{1,2,3},{4,5}} => 4
([4,1,1],5) => [3,1,1] => [[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => 4
([2,2,1],5) => [2,2,1] => [[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => 4
([3,1,1,1],5) => [2,1,1,1] => [[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => 4
([2,1,1,1,1],5) => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => 4
([6,2],5) => [4,2] => [[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => 5
([5,1,1],5) => [4,1,1] => [[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => 5
([3,3],5) => [3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => 5
([4,2,1],5) => [3,2,1] => [[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => 5
([4,1,1,1],5) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 5
([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) => [2,2,1,1] => [[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => 5
([3,1,1,1,1],5) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 5
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => {{1},{2},{3},{4},{5},{6}} => 5
([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) => [5,1] => [[1,2,3,4,5],[6]] => {{1,2,3,4,5},{6}} => 5
([4,2],6) => [4,2] => [[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => 5
([5,1,1],6) => [4,1,1] => [[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => 5
([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) => [3,1,1,1] => [[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => 5
([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) => [2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => {{1,2},{3},{4},{5},{6}} => 5
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => {{1},{2},{3},{4},{5},{6}} => 5
([7,2],6) => [5,2] => [[1,2,3,4,5],[6,7]] => {{1,2,3,4,5},{6,7}} => 6
([6,1,1],6) => [5,1,1] => [[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
([5,2,1],6) => [4,2,1] => [[1,2,3,4],[5,6],[7]] => {{1,2,3,4},{5,6},{7}} => 6
([5,1,1,1],6) => [4,1,1,1] => [[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
([4,2,1,1],6) => [3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => {{1,2,3},{4,5},{6},{7}} => 6
([4,1,1,1,1],6) => [3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => {{1,2,3},{4},{5},{6},{7}} => 6
([2,2,2,1],6) => [2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => {{1,2},{3,4},{5,6},{7}} => 6
([3,2,1,1,1],6) => [2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => {{1,2},{3,4},{5},{6},{7}} => 6
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => {{1,2},{3},{4},{5},{6},{7}} => 6
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => [[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 $P$.
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 bounded partition
Description
The (k-1)-bounded partition of a k-core.
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].
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