Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001001: 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) => [1,1] => [2] => [1,0,1,0] => 0
([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1] => [1,1,0,1,0,0] => 0
([(2,3)],4) => [1] => [1] => [1,0] => 0
([(1,3),(2,3)],4) => [1,1] => [2] => [1,0,1,0] => 0
([(0,3),(1,3),(2,3)],4) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(0,3),(1,2)],4) => [1,1] => [2] => [1,0,1,0] => 0
([(0,3),(1,2),(2,3)],4) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1] => [1,1,0,1,0,0] => 0
([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
([(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
([(3,4)],5) => [1] => [1] => [1,0] => 0
([(2,4),(3,4)],5) => [1,1] => [2] => [1,0,1,0] => 0
([(1,4),(2,4),(3,4)],5) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,4),(2,3)],5) => [1,1] => [2] => [1,0,1,0] => 0
([(1,4),(2,3),(3,4)],5) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(0,1),(2,4),(3,4)],5) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1] => [1,1,0,1,0,0] => 0
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(1,3),(1,4),(2,3),(2,4)],5) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 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) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [2,2,2] => [1,1,1,1,0,0,0,0] => 6
([(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
([(4,5)],6) => [1] => [1] => [1,0] => 0
([(3,5),(4,5)],6) => [1,1] => [2] => [1,0,1,0] => 0
([(2,5),(3,5),(4,5)],6) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(2,5),(3,4)],6) => [1,1] => [2] => [1,0,1,0] => 0
([(2,5),(3,4),(4,5)],6) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(1,2),(3,5),(4,5)],6) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1] => [1,1,0,1,0,0] => 0
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(2,4),(2,5),(3,4),(3,5)],6) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 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) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(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) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [2,2,2] => [1,1,1,1,0,0,0,0] => 6
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 0
([(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)],6) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 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] => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 0
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [2,2,2] => [1,1,1,1,0,0,0,0] => 6
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 0
([(5,6)],7) => [1] => [1] => [1,0] => 0
([(4,6),(5,6)],7) => [1,1] => [2] => [1,0,1,0] => 0
([(3,6),(4,6),(5,6)],7) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(3,6),(4,5)],7) => [1,1] => [2] => [1,0,1,0] => 0
([(3,6),(4,5),(5,6)],7) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(2,3),(4,6),(5,6)],7) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1] => [1,1,0,1,0,0] => 0
([(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,2),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(3,6),(4,5),(4,6),(5,6)],7) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(3,5),(3,6),(4,5),(4,6)],7) => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
([(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 0
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 0
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(1,6),(2,5),(3,4)],7) => [1,1,1] => [3] => [1,0,1,0,1,0] => 0
([(2,6),(3,5),(4,5),(4,6)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(1,2),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(0,3),(1,2),(4,6),(5,6)],7) => [1,1,1,1] => [4] => [1,0,1,0,1,0,1,0] => 0
([(2,3),(4,5),(4,6),(5,6)],7) => [3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 0
([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
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searching the database for the individual values of this statistic
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
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 biconnected components
Description
Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.
The biconnected components are also known as blocks of a graph.
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