Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001767: Integer partitions ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => 0
[1,2,1] => [[2,2,1],[1]] => [1] => 0
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => 1
[2,2] => [[3,2],[1]] => [1] => 0
[3,1] => [[3,3],[2]] => [2] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => 0
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 1
[1,2,2] => [[3,2,1],[1]] => [1] => 0
[1,3,1] => [[3,3,1],[2]] => [2] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 1
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => 0
[2,3] => [[4,2],[1]] => [1] => 0
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
[3,2] => [[4,3],[2]] => [2] => 1
[4,1] => [[4,4],[3]] => [3] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 1
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => 0
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 1
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 1
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 0
[1,2,3] => [[4,2,1],[1]] => [1] => 0
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
[1,3,2] => [[4,3,1],[2]] => [2] => 1
[1,4,1] => [[4,4,1],[3]] => [3] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 1
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 1
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => 0
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => 1
[2,4] => [[5,2],[1]] => [1] => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => 1
[3,3] => [[5,3],[2]] => [2] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => 1
[4,2] => [[5,4],[3]] => [3] => 1
[5,1] => [[5,5],[4]] => [4] => 2
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 0
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => 0
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 1
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => 1
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 1
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 1
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => 0
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 1
[1,2,4] => [[5,2,1],[1]] => [1] => 0
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 1
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 1
[1,3,3] => [[5,3,1],[2]] => [2] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 1
[1,4,2] => [[5,4,1],[3]] => [3] => 1
[1,5,1] => [[5,5,1],[4]] => [4] => 2
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 2
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 1
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 1
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => 0
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 1
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => 1
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => 1
[2,5] => [[6,2],[1]] => [1] => 0
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 2
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 1
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 1
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => 1
[3,4] => [[6,3],[2]] => [2] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 1
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => 1
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => 1
[4,3] => [[6,4],[3]] => [3] => 1
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => 2
[5,2] => [[6,5],[4]] => [4] => 2
[6,1] => [[6,6],[5]] => [5] => 2
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => 0
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => 1
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Description
The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Assign to each cell of the Ferrers diagram an arrow pointing north, east, south or west. Then compute for each cell the number of arrows pointing towards it, and take the minimum of those. This statistic is the maximal minimum that can be obtained by assigning arrows in any way.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
inner shape
Description
The inner shape of a skew partition.
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