Identifier
-
Mp00221:
Set partitions
—conjugate⟶
Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤ
Values
{{1}} => {{1}} => [1] => ([],1) => 0
{{1,2}} => {{1},{2}} => [1,1] => ([(0,1)],2) => 1
{{1,2,3}} => {{1},{2},{3}} => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
{{1,2},{3}} => {{1,2},{3}} => [2,1] => ([(0,2),(1,2)],3) => 1
{{1},{2,3}} => {{1,3},{2}} => [2,1] => ([(0,2),(1,2)],3) => 1
{{1,2,3,4}} => {{1},{2},{3},{4}} => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
{{1,2,3},{4}} => {{1,2},{3},{4}} => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
{{1,2,4},{3}} => {{1},{2,3},{4}} => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 1
{{1,2},{3,4}} => {{1,3},{2},{4}} => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
{{1,2},{3},{4}} => {{1,2,3},{4}} => [3,1] => ([(0,3),(1,3),(2,3)],4) => 1
{{1,4},{2,3}} => {{1},{2,4},{3}} => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 1
{{1},{2,3,4}} => {{1,4},{2},{3}} => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
{{1},{2,3},{4}} => {{1,2,4},{3}} => [3,1] => ([(0,3),(1,3),(2,3)],4) => 1
{{1},{2},{3,4}} => {{1,3,4},{2}} => [3,1] => ([(0,3),(1,3),(2,3)],4) => 1
{{1,2,3,4,5}} => {{1},{2},{3},{4},{5}} => [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) => 1
{{1,2,3,4},{5}} => {{1,2},{3},{4},{5}} => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,3,5},{4}} => {{1},{2,3},{4},{5}} => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,3},{4,5}} => {{1,3},{2},{4},{5}} => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,4,5},{3}} => {{1},{2},{3,4},{5}} => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,4},{3,5}} => {{1,3},{2,4},{5}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,4},{3},{5}} => {{1,2},{3,4},{5}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,5},{3,4}} => {{1},{2,4},{3},{5}} => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2},{3,4,5}} => {{1,4},{2},{3},{5}} => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2},{3,4},{5}} => {{1,2,4},{3},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2,5},{3},{4}} => {{1},{2,3,4},{5}} => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2},{3,5},{4}} => {{1,4},{2,3},{5}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2},{3},{4,5}} => {{1,3,4},{2},{5}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,2},{3},{4},{5}} => {{1,2,3,4},{5}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
{{1,3,4},{2,5}} => {{1,4},{2,5},{3}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,4,5},{2,3}} => {{1},{2},{3,5},{4}} => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,4},{2,3,5}} => {{1,3},{2,5},{4}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,4},{2,3},{5}} => {{1,2},{3,5},{4}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,5},{2,3,4}} => {{1},{2,5},{3},{4}} => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2,3,4,5}} => {{1,5},{2},{3},{4}} => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2,3,4},{5}} => {{1,2,5},{3},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1,5},{2,3},{4}} => {{1},{2,3,5},{4}} => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2,3,5},{4}} => {{1,5},{2,3},{4}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2,3},{4,5}} => {{1,3,5},{2},{4}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2,3},{4},{5}} => {{1,2,3,5},{4}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
{{1,5},{2},{3,4}} => {{1},{2,4,5},{3}} => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2,5},{3,4}} => {{1,5},{2,4},{3}} => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2},{3,4,5}} => {{1,4,5},{2},{3}} => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
{{1},{2},{3,4},{5}} => {{1,2,4,5},{3}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
{{1},{2},{3},{4,5}} => {{1,3,4,5},{2}} => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
{{1,2,3,4,5,6}} => {{1},{2},{3},{4},{5},{6}} => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(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) => 1
{{1,2,3,4,5},{6}} => {{1,2},{3},{4},{5},{6}} => [2,1,1,1,1] => ([(0,2),(0,3),(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) => 1
{{1,2,3,4,6},{5}} => {{1},{2,3},{4},{5},{6}} => [1,2,1,1,1] => ([(0,3),(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) => 1
{{1,2,3,4},{5,6}} => {{1,3},{2},{4},{5},{6}} => [2,1,1,1,1] => ([(0,2),(0,3),(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) => 1
{{1,2,3,4},{5},{6}} => {{1,2,3},{4},{5},{6}} => [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) => 1
{{1,2,3,5,6},{4}} => {{1},{2},{3,4},{5},{6}} => [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) => 1
{{1,2,3,5},{4,6}} => {{1,3},{2,4},{5},{6}} => [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) => 1
{{1,2,3,5},{4},{6}} => {{1,2},{3,4},{5},{6}} => [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) => 1
{{1,2,3,6},{4,5}} => {{1},{2,4},{3},{5},{6}} => [1,2,1,1,1] => ([(0,3),(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) => 1
{{1,2,3},{4,5,6}} => {{1,4},{2},{3},{5},{6}} => [2,1,1,1,1] => ([(0,2),(0,3),(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) => 1
{{1,2,3},{4,5},{6}} => {{1,2,4},{3},{5},{6}} => [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) => 1
{{1,2,3,6},{4},{5}} => {{1},{2,3,4},{5},{6}} => [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) => 1
{{1,2,3},{4,6},{5}} => {{1,4},{2,3},{5},{6}} => [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) => 1
{{1,2,3},{4},{5,6}} => {{1,3,4},{2},{5},{6}} => [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) => 1
{{1,2,3},{4},{5},{6}} => {{1,2,3,4},{5},{6}} => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,4,5,6},{3}} => {{1},{2},{3},{4,5},{6}} => [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) => 1
{{1,2,4,5},{3,6}} => {{1,4},{2,5},{3},{6}} => [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) => 1
{{1,2,4,5},{3},{6}} => {{1,2},{3},{4,5},{6}} => [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) => 1
{{1,2,4,6},{3,5}} => {{1},{2,4},{3,5},{6}} => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,4},{3,5,6}} => {{1,4},{2},{3,5},{6}} => [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) => 1
{{1,2,4},{3,5},{6}} => {{1,2,4},{3,5},{6}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,4,6},{3},{5}} => {{1},{2,3},{4,5},{6}} => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,4},{3,6},{5}} => {{1,4},{2,3,5},{6}} => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,4},{3},{5,6}} => {{1,3},{2},{4,5},{6}} => [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) => 1
{{1,2,4},{3},{5},{6}} => {{1,2,3},{4,5},{6}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,5,6},{3,4}} => {{1},{2},{3,5},{4},{6}} => [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) => 1
{{1,2,5},{3,4,6}} => {{1,3},{2,5},{4},{6}} => [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) => 1
{{1,2,5},{3,4},{6}} => {{1,2},{3,5},{4},{6}} => [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) => 1
{{1,2,6},{3,4,5}} => {{1},{2,5},{3},{4},{6}} => [1,2,1,1,1] => ([(0,3),(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) => 1
{{1,2},{3,4,5,6}} => {{1,5},{2},{3},{4},{6}} => [2,1,1,1,1] => ([(0,2),(0,3),(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) => 1
{{1,2},{3,4,5},{6}} => {{1,2,5},{3},{4},{6}} => [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) => 1
{{1,2,6},{3,4},{5}} => {{1},{2,3,5},{4},{6}} => [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) => 1
{{1,2},{3,4,6},{5}} => {{1,5},{2,3},{4},{6}} => [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) => 1
{{1,2},{3,4},{5,6}} => {{1,3,5},{2},{4},{6}} => [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) => 1
{{1,2},{3,4},{5},{6}} => {{1,2,3,5},{4},{6}} => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,5,6},{3},{4}} => {{1},{2},{3,4,5},{6}} => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,5},{3,6},{4}} => {{1,3,4},{2,5},{6}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,5},{3},{4,6}} => {{1,3},{2,4,5},{6}} => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,5},{3},{4},{6}} => {{1,2},{3,4,5},{6}} => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,6},{3,5},{4}} => {{1},{2,5},{3,4},{6}} => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2},{3,5,6},{4}} => {{1,5},{2},{3,4},{6}} => [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) => 1
{{1,2},{3,5},{4,6}} => {{1,3,5},{2,4},{6}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2},{3,5},{4},{6}} => {{1,2,5},{3,4},{6}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,6},{3},{4,5}} => {{1},{2,4,5},{3},{6}} => [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) => 1
{{1,2},{3,6},{4,5}} => {{1,5},{2,4},{3},{6}} => [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) => 1
{{1,2},{3},{4,5,6}} => {{1,4,5},{2},{3},{6}} => [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) => 1
{{1,2},{3},{4,5},{6}} => {{1,2,4,5},{3},{6}} => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2,6},{3},{4},{5}} => {{1},{2,3,4,5},{6}} => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2},{3,6},{4},{5}} => {{1,5},{2,3,4},{6}} => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2},{3},{4,6},{5}} => {{1,4,5},{2,3},{6}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2},{3},{4},{5,6}} => {{1,3,4,5},{2},{6}} => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,2},{3},{4},{5},{6}} => {{1,2,3,4,5},{6}} => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
{{1,3,4,5},{2,6}} => {{1,5},{2,6},{3},{4}} => [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) => 1
{{1,3,4,6},{2,5}} => {{1},{2,5},{3,6},{4}} => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
{{1,3,4},{2,5,6}} => {{1,5},{2},{3,6},{4}} => [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) => 1
{{1,3,4},{2,5},{6}} => {{1,2,5},{3,6},{4}} => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
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Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
This is the minimum eccentricity of any vertex.
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
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
Map
conjugate
Description
The conjugate of a set partition.
This is an involution exchanging singletons and circular adjacencies due to [1].
This is an involution exchanging singletons and circular adjacencies due to [1].
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