Identifier
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Images
=>
Cc0020;cc-rep-2Cc0020;cc-rep-3
[1]=>[1]=>([],1)=>([],1) [1,1]=>[2]=>([],2)=>([],1) [2]=>[1,1]=>([(0,1)],2)=>([(0,1)],2) [1,1,1]=>[3]=>([],3)=>([],1) [1,2]=>[1,2]=>([(1,2)],3)=>([(0,1)],2) [2,1]=>[2,1]=>([(0,2),(1,2)],3)=>([(0,1)],2) [3]=>[1,1,1]=>([(0,1),(0,2),(1,2)],3)=>([(0,1),(0,2),(1,2)],3) [1,1,1,1]=>[4]=>([],4)=>([],1) [1,1,2]=>[1,3]=>([(2,3)],4)=>([(0,1)],2) [1,2,1]=>[2,2]=>([(1,3),(2,3)],4)=>([(0,1)],2) [1,3]=>[1,1,2]=>([(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(1,2)],3) [2,1,1]=>[3,1]=>([(0,3),(1,3),(2,3)],4)=>([(0,1)],2) [2,2]=>[1,2,1]=>([(0,3),(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(1,2)],3) [3,1]=>[2,1,1]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(1,2)],3) [4]=>[1,1,1,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,1,1,1,1]=>[5]=>([],5)=>([],1) [1,1,1,2]=>[1,4]=>([(3,4)],5)=>([(0,1)],2) [1,1,2,1]=>[2,3]=>([(2,4),(3,4)],5)=>([(0,1)],2) [1,1,3]=>[1,1,3]=>([(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [1,2,1,1]=>[3,2]=>([(1,4),(2,4),(3,4)],5)=>([(0,1)],2) [1,2,2]=>[1,2,2]=>([(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [1,3,1]=>[2,1,2]=>([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [1,4]=>[1,1,1,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [2,1,1,1]=>[4,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>([(0,1)],2) [2,1,2]=>[1,3,1]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [2,2,1]=>[2,2,1]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [2,3]=>[1,1,2,1]=>([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [3,1,1]=>[3,1,1]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [3,2]=>[1,2,1,1]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [4,1]=>[2,1,1,1]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(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)=>([(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]=>[6]=>([],6)=>([],1) [1,1,1,1,2]=>[1,5]=>([(4,5)],6)=>([(0,1)],2) [1,1,1,2,1]=>[2,4]=>([(3,5),(4,5)],6)=>([(0,1)],2) [1,1,1,3]=>[1,1,4]=>([(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,1,2,1,1]=>[3,3]=>([(2,5),(3,5),(4,5)],6)=>([(0,1)],2) [1,1,2,2]=>[1,2,3]=>([(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,1,3,1]=>[2,1,3]=>([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,1,4]=>[1,1,1,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,2,1,1,1]=>[4,2]=>([(1,5),(2,5),(3,5),(4,5)],6)=>([(0,1)],2) [1,2,1,2]=>[1,3,2]=>([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,2,2,1]=>[2,2,2]=>([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,2,3]=>[1,1,2,2]=>([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,3,1,1]=>[3,1,2]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,3,2]=>[1,2,1,2]=>([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,4,1]=>[2,1,1,2]=>([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,5]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [2,1,1,1,1]=>[5,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>([(0,1)],2) [2,1,1,2]=>[1,4,1]=>([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [2,1,2,1]=>[2,3,1]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [2,1,3]=>[1,1,3,1]=>([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [2,2,1,1]=>[3,2,1]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [2,2,2]=>[1,2,2,1]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [2,3,1]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [2,4]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [3,1,1,1]=>[4,1,1]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [3,1,2]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [3,2,1]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [3,3]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [4,1,1]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [4,2]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [5,1]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(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)=>([(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)
Map
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.
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
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].