Identifier
Mp00184:
Integer compositions
—to threshold graph⟶
Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00251: Graphs —clique sizes⟶ Integer partitions
Images
=>
Cc0020;cc-rep-1Cc0002;cc-rep-2
[1]=>([],1)=>[1]
[1,1]=>([(0,1)],2)=>[2]
[2]=>([],2)=>[1,1]
[1,1,1]=>([(0,1),(0,2),(1,2)],3)=>[3]
[1,2]=>([(1,2)],3)=>[2,1]
[2,1]=>([(0,2),(1,2)],3)=>[2,2]
[3]=>([],3)=>[1,1,1]
[1,1,1,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>[4]
[1,1,2]=>([(1,2),(1,3),(2,3)],4)=>[3,1]
[1,2,1]=>([(0,3),(1,2),(1,3),(2,3)],4)=>[3,2]
[1,3]=>([(2,3)],4)=>[2,1,1]
[2,1,1]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>[3,3]
[2,2]=>([(1,3),(2,3)],4)=>[2,2,1]
[3,1]=>([(0,3),(1,3),(2,3)],4)=>[2,2,2]
[4]=>([],4)=>[1,1,1,1]
[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)=>[5]
[1,1,1,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[4,1]
[1,1,2,1]=>([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[4,2]
[1,1,3]=>([(2,3),(2,4),(3,4)],5)=>[3,1,1]
[1,2,1,1]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[4,3]
[1,2,2]=>([(1,4),(2,3),(2,4),(3,4)],5)=>[3,2,1]
[1,3,1]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>[3,2,2]
[1,4]=>([(3,4)],5)=>[2,1,1,1]
[2,1,1,1]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[4,4]
[2,1,2]=>([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[3,3,1]
[2,2,1]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[3,3,2]
[2,3]=>([(2,4),(3,4)],5)=>[2,2,1,1]
[3,1,1]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[3,3,3]
[3,2]=>([(1,4),(2,4),(3,4)],5)=>[2,2,2,1]
[4,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>[2,2,2,2]
[5]=>([],5)=>[1,1,1,1,1]
[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)=>[6]
[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)=>[5,1]
[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)=>[5,2]
[1,1,1,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1]
[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)=>[5,3]
[1,1,2,2]=>([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,2,1]
[1,1,3,1]=>([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,2,2]
[1,1,4]=>([(3,4),(3,5),(4,5)],6)=>[3,1,1,1]
[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)=>[5,4]
[1,2,1,2]=>([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,3,1]
[1,2,2,1]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,3,2]
[1,2,3]=>([(2,5),(3,4),(3,5),(4,5)],6)=>[3,2,1,1]
[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)=>[4,3,3]
[1,3,2]=>([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>[3,2,2,1]
[1,4,1]=>([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>[3,2,2,2]
[1,5]=>([(4,5)],6)=>[2,1,1,1,1]
[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)=>[5,5]
[2,1,1,2]=>([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,4,1]
[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)=>[4,4,2]
[2,1,3]=>([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,3,1,1]
[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)=>[4,4,3]
[2,2,2]=>([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,3,2,1]
[2,3,1]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,3,2,2]
[2,4]=>([(3,5),(4,5)],6)=>[2,2,1,1,1]
[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)=>[4,4,4]
[3,1,2]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,3,3,1]
[3,2,1]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,3,3,2]
[3,3]=>([(2,5),(3,5),(4,5)],6)=>[2,2,2,1,1]
[4,1,1]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,3,3,3]
[4,2]=>([(1,5),(2,5),(3,5),(4,5)],6)=>[2,2,2,2,1]
[5,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>[2,2,2,2,2]
[6]=>([],6)=>[1,1,1,1,1,1]
[1,1,1,1,1,1,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(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)=>[7]
[1,1,1,1,1,2]=>([(1,2),(1,3),(1,4),(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)=>[6,1]
[1,1,1,1,2,1]=>([(0,6),(1,2),(1,3),(1,4),(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)=>[6,2]
[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)=>[5,1,1]
[1,1,1,2,1,1]=>([(0,5),(0,6),(1,2),(1,3),(1,4),(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)=>[6,3]
[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)=>[5,2,1]
[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)=>[5,2,2]
[1,1,1,4]=>([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1]
[1,1,2,1,1,1]=>([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(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)=>[6,4]
[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)=>[5,3,1]
[1,1,2,2,1]=>([(0,6),(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)=>[5,3,2]
[1,1,2,3]=>([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,2,1,1]
[1,1,3,1,1]=>([(0,5),(0,6),(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)=>[5,3,3]
[1,1,3,2]=>([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,2,2,1]
[1,1,4,1]=>([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,2,2,2]
[1,1,5]=>([(4,5),(4,6),(5,6)],7)=>[3,1,1,1,1]
[1,2,1,1,1,1]=>([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(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)=>[6,5]
[1,2,1,1,2]=>([(1,4),(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)=>[5,4,1]
[1,2,1,2,1]=>([(0,6),(1,4),(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)=>[5,4,2]
[1,2,1,3]=>([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,3,1,1]
[1,2,2,1,1]=>([(0,5),(0,6),(1,4),(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)=>[5,4,3]
[1,2,2,2]=>([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,3,2,1]
[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)=>[4,3,2,2]
[1,2,4]=>([(3,6),(4,5),(4,6),(5,6)],7)=>[3,2,1,1,1]
[1,3,1,1,1]=>([(0,4),(0,5),(0,6),(1,4),(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)=>[5,4,4]
[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)=>[4,3,3,1]
[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)=>[4,3,3,2]
[1,3,3]=>([(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>[3,2,2,1,1]
[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)=>[4,3,3,3]
[1,4,2]=>([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>[3,2,2,2,1]
[1,5,1]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>[3,2,2,2,2]
[1,6]=>([(5,6)],7)=>[2,1,1,1,1,1]
[2,1,1,1,1,1]=>([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(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)=>[6,6]
[2,1,1,1,2]=>([(1,3),(1,4),(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)=>[5,5,1]
[2,1,1,2,1]=>([(0,6),(1,3),(1,4),(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)=>[5,5,2]
[2,1,1,3]=>([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,4,1,1]
[2,1,2,1,1]=>([(0,5),(0,6),(1,3),(1,4),(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)=>[5,5,3]
[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)=>[4,4,2,1]
[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)=>[4,4,2,2]
[2,1,4]=>([(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,1,1,1]
[2,2,1,1,1]=>([(0,4),(0,5),(0,6),(1,3),(1,4),(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)=>[5,5,4]
[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)=>[4,4,3,1]
[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)=>[4,4,3,2]
[2,2,3]=>([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,2,1,1]
[2,3,1,1]=>([(0,5),(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)=>[4,4,3,3]
[2,3,2]=>([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,2,2,1]
[2,4,1]=>([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,2,2,2]
[2,5]=>([(4,6),(5,6)],7)=>[2,2,1,1,1,1]
[3,1,1,1,1]=>([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(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)=>[5,5,5]
[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)=>[4,4,4,1]
[3,1,2,1]=>([(0,6),(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)=>[4,4,4,2]
[3,1,3]=>([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,3,1,1]
[3,2,1,1]=>([(0,5),(0,6),(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)=>[4,4,4,3]
[3,2,2]=>([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,3,2,1]
[3,3,1]=>([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,3,2,2]
[3,4]=>([(3,6),(4,6),(5,6)],7)=>[2,2,2,1,1,1]
[4,1,1,1]=>([(0,4),(0,5),(0,6),(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)=>[4,4,4,4]
[4,1,2]=>([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,3,3,1]
[4,2,1]=>([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[3,3,3,3,2]
[4,3]=>([(2,6),(3,6),(4,6),(5,6)],7)=>[2,2,2,2,1,1]
[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)=>[3,3,3,3,3]
[5,2]=>([(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[2,2,2,2,2,1]
[6,1]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[2,2,2,2,2,2]
[7]=>([],7)=>[1,1,1,1,1,1,1]
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
clique sizes
Description
The integer partition of the sizes of the maximal cliques of a graph.
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