Identifier
-
Mp00275:
Graphs
—to edge-partition of connected components⟶
Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000688: Dyck paths ⟶ ℤ
Values
([(0,1)],2) => [1] => [1] => [1,0] => 0
([(1,2)],3) => [1] => [1] => [1,0] => 0
([(0,2),(1,2)],3) => [2] => [1,1] => [1,1,0,0] => 0
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(2,3)],4) => [1] => [1] => [1,0] => 0
([(1,3),(2,3)],4) => [2] => [1,1] => [1,1,0,0] => 0
([(0,3),(1,3),(2,3)],4) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(0,3),(1,2)],4) => [1,1] => [2] => [1,0,1,0] => 0
([(0,3),(1,2),(2,3)],4) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(3,4)],5) => [1] => [1] => [1,0] => 0
([(2,4),(3,4)],5) => [2] => [1,1] => [1,1,0,0] => 0
([(1,4),(2,4),(3,4)],5) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(1,4),(2,3)],5) => [1,1] => [2] => [1,0,1,0] => 0
([(1,4),(2,3),(3,4)],5) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(0,1),(2,4),(3,4)],5) => [2,1] => [2,1] => [1,0,1,1,0,0] => 1
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(1,4),(2,3),(2,4),(3,4)],5) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(4,5)],6) => [1] => [1] => [1,0] => 0
([(3,5),(4,5)],6) => [2] => [1,1] => [1,1,0,0] => 0
([(2,5),(3,5),(4,5)],6) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(1,5),(2,5),(3,5),(4,5)],6) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(2,5),(3,4)],6) => [1,1] => [2] => [1,0,1,0] => 0
([(2,5),(3,4),(4,5)],6) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(1,2),(3,5),(4,5)],6) => [2,1] => [2,1] => [1,0,1,1,0,0] => 1
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1] => [1,1,0,1,0,0] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,1),(2,5),(3,5),(4,5)],6) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
([(2,5),(3,4),(3,5),(4,5)],6) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [2,2] => [2,2] => [1,1,1,0,0,0] => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(2,3)],6) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(1,5),(2,4),(3,4),(3,5)],6) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
([(0,1),(2,5),(3,4),(4,5)],6) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,2] => [2,2,1] => [1,1,1,0,0,1,0,0] => 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1
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Description
The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path.
The global dimension is given by St000684The global dimension of the LNakayama algebra associated to a Dyck path. and the dominant dimension is given by St000685The dominant dimension of the LNakayama algebra associated to a Dyck path.. To every Dyck path there is an LNakayama algebra associated as described in St000684The global dimension of the LNakayama algebra associated to a Dyck path..
Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least $2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].
The global dimension is given by St000684The global dimension of the LNakayama algebra associated to a Dyck path. and the dominant dimension is given by St000685The dominant dimension of the LNakayama algebra associated to a Dyck path.. To every Dyck path there is an LNakayama algebra associated as described in St000684The global dimension of the LNakayama algebra associated to a Dyck path..
Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least $2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].
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 $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
to edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.
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