Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => [1,0,1,0] => 1
[1,2,1] => [[2,2,1],[1]] => [1] => [1,0,1,0] => 1
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[2,2] => [[3,2],[1]] => [1] => [1,0,1,0] => 1
[3,1] => [[3,3],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,2,2] => [[3,2,1],[1]] => [1] => [1,0,1,0] => 1
[1,3,1] => [[3,3,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[2,3] => [[4,2],[1]] => [1] => [1,0,1,0] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[3,2] => [[4,3],[2]] => [2] => [1,1,0,0,1,0] => 1
[4,1] => [[4,4],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,2,3] => [[4,2,1],[1]] => [1] => [1,0,1,0] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,3,2] => [[4,3,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,4,1] => [[4,4,1],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[2,4] => [[5,2],[1]] => [1] => [1,0,1,0] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[3,3] => [[5,3],[2]] => [2] => [1,1,0,0,1,0] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 2
[4,2] => [[5,4],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
[5,1] => [[5,5],[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,2,4] => [[5,2,1],[1]] => [1] => [1,0,1,0] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,3,3] => [[5,3,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,4,2] => [[5,4,1],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
[1,5,1] => [[5,5,1],[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 3
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[2,5] => [[6,2],[1]] => [1] => [1,0,1,0] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[3,4] => [[6,3],[2]] => [2] => [1,1,0,0,1,0] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 2
[4,3] => [[6,4],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[5,2] => [[6,5],[4]] => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[6,1] => [[6,6],[5]] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => [1] => [1,0,1,0] => 1
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => [1,1] => [1,0,1,1,0,0] => 2
[1,1,1,3,2] => [[4,3,1,1,1],[2]] => [2] => [1,1,0,0,1,0] => 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,0,1,1,1,0,0,0] => 3
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,1,3,3] => [[5,3,1,1],[2]] => [2] => [1,1,0,0,1,0] => 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] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 3
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,3,1,3] => [[5,3,3,1],[2,2]] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,4,1,2] => [[5,4,4,1],[3,3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,4,3] => [[6,4,1],[3]] => [3] => [1,1,1,0,0,0,1,0] => 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] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 5
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => 4
[2,1,3,2] => [[5,4,2,2],[3,1,1]] => [3,1,1] => [1,0,1,1,0,0,1,0] => 3
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,3,1] => [[5,5,3,3],[4,2,2]] => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => 3
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]] => [3,3,3,3] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[4,1,1,2] => [[5,4,4,4],[3,3,3]] => [3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,1,3] => [[6,4,4],[3,3]] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 2
[4,4] => [[7,4],[3]] => [3] => [1,1,1,0,0,0,1,0] => 1
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
inner shape
Description
The inner shape of a skew partition.
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