Identifier
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Images
=>
Cc0020;cc-rep-2
[1]=>[1]=>([],1)=>[1] [1,1]=>[1,1]=>([(0,1)],2)=>[1,1] [2]=>[2]=>([],2)=>[2] [1,1,1]=>[1,1,1]=>([(0,1),(0,2),(1,2)],3)=>[1,1,1] [1,2]=>[2,1]=>([(0,2),(1,2)],3)=>[2,1] [2,1]=>[1,2]=>([(1,2)],3)=>[2,1] [3]=>[3]=>([],3)=>[3] [1,1,1,1]=>[1,1,1,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>[1,1,1,1] [1,1,2]=>[1,2,1]=>([(0,3),(1,2),(1,3),(2,3)],4)=>[2,1,1] [1,2,1]=>[2,1,1]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>[2,1,1] [1,3]=>[3,1]=>([(0,3),(1,3),(2,3)],4)=>[3,1] [2,1,1]=>[1,1,2]=>([(1,2),(1,3),(2,3)],4)=>[2,1,1] [2,2]=>[2,2]=>([(1,3),(2,3)],4)=>[3,1] [3,1]=>[1,3]=>([(2,3)],4)=>[3,1] [4]=>[4]=>([],4)=>[4] [1,1,1,1,1]=>[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)=>[1,1,1,1,1] [1,1,1,2]=>[1,1,2,1]=>([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[2,1,1,1] [1,1,2,1]=>[1,2,1,1]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[2,1,1,1] [1,1,3]=>[1,3,1]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>[3,1,1] [1,2,1,1]=>[2,1,1,1]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[2,1,1,1] [1,2,2]=>[2,2,1]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[3,1,1] [1,3,1]=>[3,1,1]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[3,1,1] [1,4]=>[4,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>[4,1] [2,1,1,1]=>[1,1,1,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[2,1,1,1] [2,1,2]=>[1,2,2]=>([(1,4),(2,3),(2,4),(3,4)],5)=>[3,1,1] [2,2,1]=>[2,1,2]=>([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>[3,1,1] [2,3]=>[3,2]=>([(1,4),(2,4),(3,4)],5)=>[4,1] [3,1,1]=>[1,1,3]=>([(2,3),(2,4),(3,4)],5)=>[3,1,1] [3,2]=>[2,3]=>([(2,4),(3,4)],5)=>[4,1] [4,1]=>[1,4]=>([(3,4)],5)=>[4,1] [5]=>[5]=>([],5)=>[5] [1,1,1,1,1,1]=>[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)=>[1,1,1,1,1,1] [1,1,1,1,2]=>[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)=>[2,1,1,1,1] [1,1,1,2,1]=>[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)=>[2,1,1,1,1] [1,1,1,3]=>[1,1,3,1]=>([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,1,1,1] [1,1,2,1,1]=>[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)=>[2,1,1,1,1] [1,1,2,2]=>[1,2,2,1]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,1,1,1] [1,1,3,1]=>[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)=>[3,1,1,1] [1,1,4]=>[1,4,1]=>([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [1,2,1,1,1]=>[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)=>[2,1,1,1,1] [1,2,1,2]=>[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)=>[3,1,1,1] [1,2,2,1]=>[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)=>[3,1,1,1] [1,2,3]=>[2,3,1]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [1,3,1,1]=>[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)=>[3,1,1,1] [1,3,2]=>[3,2,1]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [1,4,1]=>[4,1,1]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [1,5]=>[5,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>[5,1] [2,1,1,1,1]=>[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)=>[2,1,1,1,1] [2,1,1,2]=>[1,1,2,2]=>([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,1,1,1] [2,1,2,1]=>[1,2,1,2]=>([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,1,1,1] [2,1,3]=>[1,3,2]=>([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [2,2,1,1]=>[2,1,1,2]=>([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,1,1,1] [2,2,2]=>[2,2,2]=>([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [2,3,1]=>[3,1,2]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [2,4]=>[4,2]=>([(1,5),(2,5),(3,5),(4,5)],6)=>[5,1] [3,1,1,1]=>[1,1,1,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[3,1,1,1] [3,1,2]=>[1,2,3]=>([(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [3,2,1]=>[2,1,3]=>([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>[4,1,1] [3,3]=>[3,3]=>([(2,5),(3,5),(4,5)],6)=>[5,1] [4,1,1]=>[1,1,4]=>([(3,4),(3,5),(4,5)],6)=>[4,1,1] [4,2]=>[2,4]=>([(3,5),(4,5)],6)=>[5,1] [5,1]=>[1,5]=>([(4,5)],6)=>[5,1] [6]=>[6]=>([],6)=>[6] [1,1,1,1,1,1,1]=>[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)=>[1,1,1,1,1,1,1] [1,1,1,1,1,2]=>[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)=>[2,1,1,1,1,1] [1,1,1,1,2,1]=>[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)=>[2,1,1,1,1,1] [1,1,1,1,3]=>[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)=>[3,1,1,1,1] [1,1,1,2,1,1]=>[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)=>[2,1,1,1,1,1] [1,1,1,2,2]=>[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)=>[3,1,1,1,1] [1,1,1,3,1]=>[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)=>[3,1,1,1,1] [1,1,1,4]=>[1,1,4,1]=>([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [1,1,2,1,1,1]=>[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)=>[2,1,1,1,1,1] [1,1,2,1,2]=>[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)=>[3,1,1,1,1] [1,1,2,2,1]=>[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)=>[3,1,1,1,1] [1,1,2,3]=>[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)=>[4,1,1,1] [1,1,3,1,1]=>[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)=>[3,1,1,1,1] [1,1,3,2]=>[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)=>[4,1,1,1] [1,1,4,1]=>[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)=>[4,1,1,1] [1,1,5]=>[1,5,1]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [1,2,1,1,1,1]=>[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)=>[2,1,1,1,1,1] [1,2,1,1,2]=>[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)=>[3,1,1,1,1] [1,2,1,2,1]=>[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)=>[3,1,1,1,1] [1,2,1,3]=>[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)=>[4,1,1,1] [1,2,2,1,1]=>[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)=>[3,1,1,1,1] [1,2,2,2]=>[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)=>[4,1,1,1] [1,2,3,1]=>[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)=>[4,1,1,1] [1,2,4]=>[2,4,1]=>([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [1,3,1,1,1]=>[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)=>[3,1,1,1,1] [1,3,1,2]=>[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)=>[4,1,1,1] [1,3,2,1]=>[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)=>[4,1,1,1] [1,3,3]=>[3,3,1]=>([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [1,4,1,1]=>[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)=>[4,1,1,1] [1,4,2]=>[4,2,1]=>([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [1,5,1]=>[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)=>[5,1,1] [1,6]=>[6,1]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[6,1] [2,1,1,1,1,1]=>[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)=>[2,1,1,1,1,1] [2,1,1,1,2]=>[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)=>[3,1,1,1,1] [2,1,1,2,1]=>[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)=>[3,1,1,1,1] [2,1,1,3]=>[1,1,3,2]=>([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [2,1,2,1,1]=>[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)=>[3,1,1,1,1] [2,1,2,2]=>[1,2,2,2]=>([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [2,1,3,1]=>[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)=>[4,1,1,1] [2,1,4]=>[1,4,2]=>([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [2,2,1,1,1]=>[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)=>[3,1,1,1,1] [2,2,1,2]=>[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)=>[4,1,1,1] [2,2,2,1]=>[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)=>[4,1,1,1] [2,2,3]=>[2,3,2]=>([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [2,3,1,1]=>[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)=>[4,1,1,1] [2,3,2]=>[3,2,2]=>([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [2,4,1]=>[4,1,2]=>([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [2,5]=>[5,2]=>([(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>[6,1] [3,1,1,1,1]=>[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)=>[3,1,1,1,1] [3,1,1,2]=>[1,1,2,3]=>([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [3,1,2,1]=>[1,2,1,3]=>([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [3,1,3]=>[1,3,3]=>([(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [3,2,1,1]=>[2,1,1,3]=>([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [3,2,2]=>[2,2,3]=>([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [3,3,1]=>[3,1,3]=>([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [3,4]=>[4,3]=>([(2,6),(3,6),(4,6),(5,6)],7)=>[6,1] [4,1,1,1]=>[1,1,1,4]=>([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[4,1,1,1] [4,1,2]=>[1,2,4]=>([(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [4,2,1]=>[2,1,4]=>([(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>[5,1,1] [4,3]=>[3,4]=>([(3,6),(4,6),(5,6)],7)=>[6,1] [5,1,1]=>[1,1,5]=>([(4,5),(4,6),(5,6)],7)=>[5,1,1] [5,2]=>[2,5]=>([(4,6),(5,6)],7)=>[6,1] [6,1]=>[1,6]=>([(5,6)],7)=>[6,1] [7]=>[7]=>([],7)=>[7]
Map
rotate front to back
Description
The front to back rotation of the entries of an integer composition.
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.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.