Identifier
-
Mp00276:
Graphs
—to edge-partition of biconnected components⟶
Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001605: Integer partitions ⟶ ℤ
Values
([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,4),(1,4),(2,3),(3,4)],5) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,4),(1,3),(2,3),(2,4)],5) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [3,3] => [3] => [1,1,1] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,1),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,4),(3,4)],6) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(1,5),(2,4),(3,4),(3,5)],6) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,1),(2,5),(3,4),(4,5)],6) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3] => [3] => [1,1,1] => 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [3,3,1] => [3,1] => [2,1,1] => 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => [4,3] => [3] => [1,1,1] => 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,3,1] => [3,1] => [2,1,1] => 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,3] => [3] => [1,1,1] => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => [3,3] => [3] => [1,1,1] => 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => [3,3,1] => [3,1] => [2,1,1] => 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [5,3] => [3] => [1,1,1] => 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,3] => [3] => [1,1,1] => 2
([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(1,2),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [4,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(2,6),(3,5),(4,5),(4,6)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(1,2),(3,6),(4,5),(5,6)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(0,3),(1,2),(4,6),(5,6)],7) => [1,1,1,1] => [1,1,1] => [3] => 1
([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3] => [3] => [1,1,1] => 2
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1] => [3,1] => [2,1,1] => 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => [3,3,1,1] => [3,1,1] => [3,1,1] => 4
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => [4,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [1,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [1,1,1,1,1] => [5] => 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => [3] => 1
([(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [4,3] => [3] => [1,1,1] => 2
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,1] => [3,1] => [2,1,1] => 3
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7) => [4,3,1] => [3,1] => [2,1,1] => 3
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3] => [3] => [1,1,1] => 2
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,3,1,1] => [3,1,1] => [3,1,1] => 4
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,3,1] => [3,1] => [2,1,1] => 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) => [3,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => [4] => 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [1,1,1] => [3] => 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => [4,1,1,1] => [1,1,1] => [3] => 1
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Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
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
first row removal
Description
Removes the first entry of an integer partition
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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