Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000053: Dyck paths ⟶ ℤ (values match St000015The number of peaks of a Dyck path., St001068Number of torsionless simple modules in the corresponding Nakayama algebra., St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.)
Values
[2,1] => [[2,2],[1]] => [1] => [1,0] => 0
[1,2,1] => [[2,2,1],[1]] => [1] => [1,0] => 0
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0
[2,2] => [[3,2],[1]] => [1] => [1,0] => 0
[3,1] => [[3,3],[2]] => [2] => [1,0,1,0] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => [1,0] => 0
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0
[1,2,2] => [[3,2,1],[1]] => [1] => [1,0] => 0
[1,3,1] => [[3,3,1],[2]] => [2] => [1,0,1,0] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[2,3] => [[4,2],[1]] => [1] => [1,0] => 0
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0
[3,2] => [[4,3],[2]] => [2] => [1,0,1,0] => 1
[4,1] => [[4,4],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => [1,1,0,0] => 0
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => [1,0] => 0
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => [1,0,1,0] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[1,2,3] => [[4,2,1],[1]] => [1] => [1,0] => 0
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0
[1,3,2] => [[4,3,1],[2]] => [2] => [1,0,1,0] => 1
[1,4,1] => [[4,4,1],[3]] => [3] => [1,0,1,0,1,0] => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[2,4] => [[5,2],[1]] => [1] => [1,0] => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[3,3] => [[5,3],[2]] => [2] => [1,0,1,0] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[4,2] => [[5,4],[3]] => [3] => [1,0,1,0,1,0] => 2
[5,1] => [[5,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => [1,1,0,0] => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => [1,0,1,0] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => [1,1,0,0] => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => [1,0] => 0
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => [1,0,1,0] => 1
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => [1,1,0,0] => 0
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[1,2,4] => [[5,2,1],[1]] => [1] => [1,0] => 0
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[1,3,3] => [[5,3,1],[2]] => [2] => [1,0,1,0] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[1,4,2] => [[5,4,1],[3]] => [3] => [1,0,1,0,1,0] => 2
[1,5,1] => [[5,5,1],[4]] => [4] => [1,0,1,0,1,0,1,0] => 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 3
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1,0,1,0,0] => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => [1,1,0,0] => 0
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => [1,0,1,1,0,0] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 2
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => [1,0,1,0,1,1,0,0] => 2
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[2,5] => [[6,2],[1]] => [1] => [1,0] => 0
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [1,1,1,1,0,0,0,0] => 0
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [1,1,1,0,0,0] => 0
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 2
[3,4] => [[6,3],[2]] => [2] => [1,0,1,0] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 0
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[4,3] => [[6,4],[3]] => [3] => [1,0,1,0,1,0] => 2
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 2
[5,2] => [[6,5],[4]] => [4] => [1,0,1,0,1,0,1,0] => 3
[6,1] => [[6,6],[5]] => [5] => [1,0,1,0,1,0,1,0,1,0] => 4
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => [1,0] => 0
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => [1,1,0,0] => 0
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Description
The number of valleys of the Dyck path.
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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
inner shape
Description
The inner shape of a skew partition.
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