Identifier
-
Mp00043:
Integer partitions
—to Dyck path⟶
Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[1] => [1,0,1,0] => [1,1,0,0] => [1,0,1,0] => 1
[2] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 2
[3] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 2
[2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 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] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[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] => [1,1,0,1,1,0,0,0,1,1,0,0] => 6
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,1,0,0] => 8
[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] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[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] => [1,0,1,1,1,0,0,1,1,0,0,0] => 7
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 5
[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] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0] => 8
[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] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 5
[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] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3
[6,2,2] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0] => 6
[6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => 8
[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] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[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] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[6,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[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] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5
[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] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[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] => [1,0,1,1,1,1,0,0,0,1,0,0] => 6
[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] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[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] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[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] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[6,4,1,1] => [1,1,1,0,1,1,0,0,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,1,1,0,0,0] => 7
[6,2,2,2] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,1,1,0,0,0] => 7
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 7
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[4,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 6
[6,4,3] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[6,4,1,1,1] => [1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => 9
[6,2,2,2,1] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => 10
[6,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9
[5,5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 5
[4,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 6
[6,4,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => 5
[6,2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 7
[6,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => 10
[5,5,4] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 4
[5,5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5
[5,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => 8
[5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[6,4,3,2] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[6,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,1,0,0] => 6
[6,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => 7
[6,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => 9
[5,5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6
[4,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 5
[6,4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0] => 7
[6,4,3,2,1] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[6,3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 8
[5,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5
[5,5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[5,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 3
[5,3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0] => 7
[5,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4
[4,4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6
[6,4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 9
[6,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0] => 5
[6,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0,1,0] => 7
[5,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0,1,0] => 7
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
swap returns and last descent
Description
Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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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