Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001913: Integer partitions ⟶ ℤ
Values
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => [1] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => [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
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => [1] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [2] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => [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
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => [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] => 0
[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
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => [1] => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 2
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => [1] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => [1] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 1
[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] => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => [1] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[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
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => [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] => 0
[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
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => [1] => 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 2
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => [1] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => [1] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 1
[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] => 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] => 0
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [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] => [1,1,1] => 0
[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] => 1
[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
[2,1,4] => [[5,2,2],[1,1]] => [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] => 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 2
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => [1] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 2
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => [1] => 1
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => [1] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [2,2,2] => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 1
[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] => 2
[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] => 1
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [3] => 1
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [3] => 1
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [4] => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => [1] => 1
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => [1,1] => [1] => 1
[1,1,1,2,2,1] => [[3,3,2,1,1,1],[2,1]] => [2,1] => [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] => 0
[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] => 2
[1,1,2,2,2] => [[4,3,2,1,1],[2,1]] => [2,1] => [1] => 1
[1,1,2,3,1] => [[4,4,2,1,1],[3,1]] => [3,1] => [1] => 1
[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] => 0
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 0
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 0
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 1
[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] => 1
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 2
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => [3,2,1] => [2,1] => 2
[1,2,2,3] => [[5,3,2,1],[2,1]] => [2,1] => [1] => 1
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => [3,3,1] => [3,1] => 2
[1,2,3,2] => [[5,4,2,1],[3,1]] => [3,1] => [1] => 1
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 1
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => [3,2,2] => [2,2] => 1
[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] => 2
[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] => 1
[1,4,1,2] => [[5,4,4,1],[3,3]] => [3,3] => [3] => 1
[1,4,2,1] => [[5,5,4,1],[4,3]] => [4,3] => [3] => 1
[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] => 0
[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] => 0
[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] => 0
[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] => 0
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Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
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
first row removal
Description
Removes the first entry of an integer partition
Map
inner shape
Description
The inner shape of a skew partition.
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