Identifier
-
Mp00184:
Integer compositions
—to threshold graph⟶
Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000481: Integer partitions ⟶ ℤ
Values
[1] => ([],1) => [] => [] => 0
[1,1] => ([(0,1)],2) => [1] => [1] => 0
[2] => ([],2) => [] => [] => 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1] => 1
[1,2] => ([(1,2)],3) => [1] => [1] => 0
[2,1] => ([(0,2),(1,2)],3) => [1,1] => [2] => 0
[3] => ([],3) => [] => [] => 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => [1,1,1,1,1,1] => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => [3] => [1,1,1] => 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => [2,1,1] => 1
[1,3] => ([(2,3)],4) => [1] => [1] => 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => [1,1,1,1,1] => 1
[2,2] => ([(1,3),(2,3)],4) => [1,1] => [2] => 0
[3,1] => ([(0,3),(1,3),(2,3)],4) => [1,1,1] => [3] => 0
[4] => ([],4) => [] => [] => 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => [1,1,1,1,1,1] => 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6,1] => [2,1,1,1,1,1] => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => [3] => [1,1,1] => 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [8] => [1,1,1,1,1,1,1,1] => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => [2,1,1] => 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => [3,1,1] => 1
[1,4] => ([(3,4)],5) => [1] => [1] => 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [9] => [1,1,1,1,1,1,1,1,1] => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1] => 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => [2,1,1,1,1] => 1
[2,3] => ([(2,4),(3,4)],5) => [1,1] => [2] => 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => [1,1,1,1,1,1,1] => 1
[3,2] => ([(1,4),(2,4),(3,4)],5) => [1,1,1] => [3] => 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => [4] => 0
[5] => ([],5) => [] => [] => 0
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,1,1,1,1,1] => 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => [2,1,1,1,1,1] => 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1,1] => [3,1,1,1,1,1] => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => [3] => [1,1,1] => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8] => [1,1,1,1,1,1,1,1] => 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8,1] => [2,1,1,1,1,1,1,1] => 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => [3,1] => [2,1,1] => 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1] => [3,1,1] => 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [4,1,1] => 1
[1,5] => ([(4,5)],6) => [1] => [1] => 0
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9,1] => [2,1,1,1,1,1,1,1,1] => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => [1,1,1,1,1] => 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [11] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [2,1,1,1,1] => 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1,1] => [3,1,1,1,1] => 1
[2,4] => ([(3,5),(4,5)],6) => [1,1] => [2] => 0
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => [1,1,1,1,1,1,1] => 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7,1] => [2,1,1,1,1,1,1] => 1
[3,3] => ([(2,5),(3,5),(4,5)],6) => [1,1,1] => [3] => 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1] => [4] => 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => [5] => 0
[6] => ([],6) => [] => [] => 0
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [10,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [10,1,1] => [3,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6] => [1,1,1,1,1,1] => 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [2,1,1,1,1,1] => 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1,1] => [3,1,1,1,1,1] => 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1,1,1] => [4,1,1,1,1,1] => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => [3] => [1,1,1] => 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8] => [1,1,1,1,1,1,1,1] => 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8,1] => [2,1,1,1,1,1,1,1] => 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8,1,1] => [3,1,1,1,1,1,1,1] => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => [3,1] => [2,1,1] => 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [10,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1] => [3,1,1] => 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1] => [4,1,1] => 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [5,1,1] => 1
[1,6] => ([(5,6)],7) => [1] => [1] => 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [9] => [1,1,1,1,1,1,1,1,1] => 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [9,1] => [2,1,1,1,1,1,1,1,1] => 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [9,1,1] => [3,1,1,1,1,1,1,1,1] => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => [1,1,1,1,1] => 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [11] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [11,1] => [2,1,1,1,1,1,1,1,1,1,1] => 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1] => [2,1,1,1,1] => 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [3,1,1,1,1] => 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1,1] => [4,1,1,1,1] => 1
[2,5] => ([(4,6),(5,6)],7) => [1,1] => [2] => 0
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7] => [1,1,1,1,1,1,1] => 1
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7,1] => [2,1,1,1,1,1,1] => 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7,1,1] => [3,1,1,1,1,1,1] => 1
[3,4] => ([(3,6),(4,6),(5,6)],7) => [1,1,1] => [3] => 0
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [9] => [1,1,1,1,1,1,1,1,1] => 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [9,1] => [2,1,1,1,1,1,1,1,1] => 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1] => [4] => 0
[5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [11] => [1,1,1,1,1,1,1,1,1,1,1] => 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1] => [5] => 0
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [1,1,1,1,1,1] => [6] => 0
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Description
The number of upper covers of a partition in dominance order.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
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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