Identifier
-
Mp00044:
Integer partitions
—conjugate⟶
Integer partitions
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001195: Dyck paths ⟶ ℤ
Values
[3] => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
[2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 1
[1,1,1] => [3] => [2,1] => [1,0,1,1,0,0] => 1
[4] => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
[3,1] => [2,1,1] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[2,2] => [2,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => 1
[2,1,1] => [3,1] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
[1,1,1,1] => [4] => [2,2] => [1,1,1,0,0,0] => 1
[5] => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[4,1] => [2,1,1,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[3,2] => [2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1
[3,1,1] => [3,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1
[2,2,1] => [3,2] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 1
[2,1,1,1] => [4,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1
[1,1,1,1,1] => [5] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[4,2] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[3,3] => [2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => 1
[3,1,1,1] => [4,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1
[2,1,1,1,1] => [5,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[1,1,1,1,1,1] => [6] => [3,3] => [1,1,1,0,1,0,0,0] => 1
[4,3] => [2,2,2,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1
[4,1,1,1] => [4,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 1
[2,1,1,1,1,1] => [6,1] => [3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1
[1,1,1,1,1,1,1] => [7] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[4,4] => [2,2,2,2] => [3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1
[4,1,1,1,1] => [5,1,1,1] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,1,1,1,1,1,1] => [8] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
[5,4] => [2,2,2,2,1] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 1
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Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
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.
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.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition λ of n is the partition λ∗ whose Ferrers diagram is obtained from the diagram of λ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition λ of n is the partition λ∗ whose Ferrers diagram is obtained from the diagram of λ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
Loehr-Warrington inverse
Description
Return a partition whose length is the diagonal inversion number of the preimage.
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