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 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
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
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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