Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => 1
[1,2,1] => [[2,2,1],[1]] => [1] => 1
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => 0
[2,2] => [[3,2],[1]] => [1] => 1
[3,1] => [[3,3],[2]] => [2] => 0
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 0
[1,2,2] => [[3,2,1],[1]] => [1] => 1
[1,3,1] => [[3,3,1],[2]] => [2] => 0
[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] => 0
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => 1
[2,3] => [[4,2],[1]] => [1] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => 2
[3,2] => [[4,3],[2]] => [2] => 0
[4,1] => [[4,4],[3]] => [3] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 0
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => 1
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => 0
[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] => 0
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 1
[1,2,3] => [[4,2,1],[1]] => [1] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 2
[1,3,2] => [[4,3,1],[2]] => [2] => 0
[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] => 0
[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] => 0
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => 0
[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] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => 0
[2,4] => [[5,2],[1]] => [1] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 2
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => 2
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => 1
[3,3] => [[5,3],[2]] => [2] => 0
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => 2
[4,2] => [[5,4],[3]] => [3] => 1
[5,1] => [[5,5],[4]] => [4] => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 0
[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] => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 2
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => 0
[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] => 0
[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] => 0
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 0
[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] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 0
[1,2,4] => [[5,2,1],[1]] => [1] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 2
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 2
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 1
[1,3,3] => [[5,3,1],[2]] => [2] => 0
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 2
[1,4,2] => [[5,4,1],[3]] => [3] => 1
[1,5,1] => [[5,5,1],[4]] => [4] => 0
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[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] => 0
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 0
[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] => 0
[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] => 0
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 3
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => 0
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => 1
[2,5] => [[6,2],[1]] => [1] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 0
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 3
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 0
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => 0
[3,4] => [[6,3],[2]] => [2] => 0
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 2
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => 2
[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] => 0
[5,2] => [[6,5],[4]] => [4] => 0
[6,1] => [[6,6],[5]] => [5] => 1
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => 0
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Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
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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