Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
St001966: Dyck paths ⟶ ℤ
Values
[1] => [1,0,1,0] => 4
[2] => [1,1,0,0,1,0] => 4
[1,1] => [1,0,1,1,0,0] => 4
[3] => [1,1,1,0,0,0,1,0] => 4
[2,1] => [1,0,1,0,1,0] => 7
[1,1,1] => [1,0,1,1,1,0,0,0] => 4
[4] => [1,1,1,1,0,0,0,0,1,0] => 4
[3,1] => [1,1,0,1,0,0,1,0] => 7
[2,2] => [1,1,0,0,1,1,0,0] => 4
[2,1,1] => [1,0,1,1,0,1,0,0] => 7
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 4
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 4
[4,1] => [1,1,1,0,1,0,0,0,1,0] => 7
[3,2] => [1,1,0,0,1,0,1,0] => 7
[3,1,1] => [1,0,1,1,0,0,1,0] => 7
[2,2,1] => [1,0,1,0,1,1,0,0] => 7
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 7
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 4
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 7
[4,2] => [1,1,1,0,0,1,0,0,1,0] => 7
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 7
[3,3] => [1,1,1,0,0,0,1,1,0,0] => 4
[3,2,1] => [1,0,1,0,1,0,1,0] => 10
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 7
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 4
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 7
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 7
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => 7
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[4,3] => [1,1,1,0,0,0,1,0,1,0] => 7
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 7
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 7
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 7
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 7
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 7
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 7
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 7
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => 7
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => 7
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => 7
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => 10
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => 7
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => 10
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 7
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => 7
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => 7
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => 10
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => 7
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => 7
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => 7
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => 7
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => 7
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => 7
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 7
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => 7
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => 10
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => 10
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => 10
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 7
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 4
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => 10
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => 7
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => 7
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => 7
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => 7
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => 10
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => 7
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => 10
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => 7
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => 10
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => 7
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => 7
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => 7
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => 13
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 7
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => 7
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => 10
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => 7
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => 7
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => 7
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => 10
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => 10
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => 10
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => 7
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => 10
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => 10
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => 10
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => 7
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => 7
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => 10
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => 7
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 10
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 10
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => 7
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => 7
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => 10
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Description
Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path).
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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