Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001219: Dyck paths ⟶ ℤ
Values
[1] => [1,0,1,0] => [1,1,0,0] => [1,0,1,0] => 0
[2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 0
[1,1] => [1,0,1,1,0,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 1
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 0
[2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 3
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 4
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 0
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 0
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1
[3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 2
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 0
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 1
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 2
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 3
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 0
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 0
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 0
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 1
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 2
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 0
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 0
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 0
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 0
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 0
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 2
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 2
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 2
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 3
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 0
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 0
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 0
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 0
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 1
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 1
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 0
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 1
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 2
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 2
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 0
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 0
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 0
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 0
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 0
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 0
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 1
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 1
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 1
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 1
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 1
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 1
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 2
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 2
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 2
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Description
Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength 0 is sent to itself.
Let D be a Dyck path of semilength n>0 and decompose it into 1D10D2 with Dyck paths D1,D2 of respective semilengths n1 and n2 such that n1 is minimal. One then has n1+n2=n−1.
Now let ˜D1 and ˜D2 be the recursively defined respective images of D1 and D2 under this map. The image of D is then defined as 1˜D20˜D1.
The unique empty path of semilength 0 is sent to itself.
Let D be a Dyck path of semilength n>0 and decompose it into 1D10D2 with Dyck paths D1,D2 of respective semilengths n1 and n2 such that n1 is minimal. One then has n1+n2=n−1.
Now let ˜D1 and ˜D2 be the recursively defined respective images of D1 and D2 under this map. The image of D is then defined as 1˜D20˜D1.
Map
inverse promotion
Description
The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
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