Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤ
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [2] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [2] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[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],[2,1,1]] => [2,1,1] => [1,1] => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 2
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [2] => 1
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [2] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [3] => 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [2] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [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] => 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [1,1] => 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 0
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 2
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [2] => 1
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [2] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [3] => 2
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [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] => [1,1,1] => 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,1] => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 0
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 0
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 0
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 3
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [2,2,2] => 4
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 2
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [2] => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [3,2] => 0
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [2] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [2] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [3,3] => 4
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [3] => 2
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [3] => 2
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [4] => 3
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => [2,1,1] => [1,1] => 1
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => [2,2,1] => [2,1] => 0
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => [2,2] => [2] => 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => [3,2] => [2] => 1
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 3
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 2
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 3
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]] => [2,1,1] => [1,1] => 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => [3,1,1] => [1,1] => 1
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 0
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 0
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => [3,2,1] => [2,1] => 0
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => [3,3,1] => [3,1] => 3
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 2
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => [3,2,2] => [2,2] => 2
[1,3,1,3] => [[5,3,3,1],[2,2]] => [2,2] => [2] => 1
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => [3,3,2] => [3,2] => 0
[1,3,2,2] => [[5,4,3,1],[3,2]] => [3,2] => [2] => 1
[1,3,3,1] => [[5,5,3,1],[4,2]] => [4,2] => [2] => 1
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => [3,3,3] => [3,3] => 4
[1,4,1,2] => [[5,4,4,1],[3,3]] => [3,3] => [3] => 2
[1,4,2,1] => [[5,5,4,1],[4,3]] => [4,3] => [3] => 2
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [1,1,1,1,1,1] => [1,1,1,1,1] => 4
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => [2,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => [2,2,1,1,1] => [2,1,1,1] => 2
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 2
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => [3,1,1,1] => [1,1,1] => 2
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => [2,2,2,1,1] => [2,2,1,1] => 4
[2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 3
[2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]] => [3,2,1,1] => [2,1,1] => 3
[2,1,2,3] => [[5,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 1
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => [3,3,1,1] => [3,1,1] => 0
[2,1,3,2] => [[5,4,2,2],[3,1,1]] => [3,1,1] => [1,1] => 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => [2,2,2,2,1] => [2,2,2,1] => 0
[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 0
[2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]] => [3,2,2,1] => [2,2,1] => 0
[2,2,1,3] => [[5,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 0
[2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]] => [3,3,2,1] => [3,2,1] => 1
[2,2,2,2] => [[5,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 0
[2,2,3,1] => [[5,5,3,2],[4,2,1]] => [4,2,1] => [2,1] => 0
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => [3,3,3,1] => [3,3,1] => 3
[2,3,1,2] => [[5,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 3
[2,3,2,1] => [[5,5,4,2],[4,3,1]] => [4,3,1] => [3,1] => 3
[2,4,1,1] => [[5,5,5,2],[4,4,1]] => [4,4,1] => [4,1] => 2
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [2,2,2,2,2] => [2,2,2,2] => 5
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => [3,2,2,2] => [2,2,2] => 4
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 2
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => [3,3,2,2] => [3,2,2] => 3
[3,1,2,2] => [[5,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 2
[3,1,3,1] => [[5,5,3,3],[4,2,2]] => [4,2,2] => [2,2] => 2
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => [3,3,3,2] => [3,3,2] => 1
[3,2,1,2] => [[5,4,4,3],[3,3,2]] => [3,3,2] => [3,2] => 0
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Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Map
first row removal
Description
Removes the first entry of an integer partition
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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