Identifier
  • Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
    St001007: Dyck paths ⟶ ℤ (values match St000024The number of double up and double down steps of a Dyck path., St000443The number of long tunnels of a Dyck path., St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra., St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.)
Values
[1] => [1,0] => 1
[2] => [1,0,1,0] => 1
[1,1] => [1,1,0,0] => 2
[3] => [1,0,1,0,1,0] => 1
[2,1] => [1,0,1,1,0,0] => 2
[1,1,1] => [1,1,0,1,0,0] => 2
[4] => [1,0,1,0,1,0,1,0] => 1
[3,1] => [1,0,1,0,1,1,0,0] => 2
[2,2] => [1,1,1,0,0,0] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => 2
[5] => [1,0,1,0,1,0,1,0,1,0] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => 2
[3,2] => [1,0,1,1,1,0,0,0] => 3
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => 3
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 2
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 2
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[4,2] => [1,0,1,0,1,1,1,0,0,0] => 3
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 2
[3,3] => [1,1,1,0,1,0,0,0] => 3
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 3
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => 4
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 2
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 2
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 3
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 3
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 4
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 3
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 2
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 4
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 2
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 2
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 3
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => 3
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 3
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 4
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 4
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 3
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 4
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 4
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 3
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 3
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 3
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 3
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 4
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 3
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 5
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 4
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 3
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 4
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 4
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 4
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 4
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 3
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 3
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 3
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 4
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => 4
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 3
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 4
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 5
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => 4
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 4
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 4
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => 4
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 4
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 3
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 3
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 4
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 5
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 4
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 5
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 4
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 5
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 5
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 4
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 4
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 4
>>> Load all 127 entries. <<<
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 3
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 4
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 5
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 5
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 4
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 5
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 5
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 4
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 4
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 5
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 5
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 5
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 5
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 6
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 5
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 5
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 5
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 6
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 6
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 5
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 6
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 6
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
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Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
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.