Identifier
Mp00319: Parking functions —to composition⟶ 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]=>[1,1]=>([(0,1)],2)=>([(0,1)],2) [1,2]=>[1,2]=>([(1,2)],3)=>([(0,1)],2) [2,1]=>[2,1]=>([(0,2),(1,2)],3)=>([(0,1)],2) [1,1,1]=>[1,1,1]=>([(0,1),(0,2),(1,2)],3)=>([(0,1),(0,2),(1,2)],3) [1,1,2]=>[1,1,2]=>([(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(1,2)],3) [1,2,1]=>[1,2,1]=>([(0,3),(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(1,2)],3) [2,1,1]=>[2,1,1]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>([(0,1),(0,2),(1,2)],3) [1,1,3]=>[1,1,3]=>([(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [1,3,1]=>[1,3,1]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [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) [1,2,2]=>[1,2,2]=>([(1,4),(2,3),(2,4),(3,4)],5)=>([(0,1),(0,2),(1,2)],3) [2,1,2]=>[2,1,2]=>([(1,3),(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) [1,2,3]=>[1,2,3]=>([(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [1,3,2]=>[1,3,2]=>([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [2,1,3]=>[2,1,3]=>([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>([(0,1),(0,2),(1,2)],3) [2,3,1]=>[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) [3,1,2]=>[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) [3,2,1]=>[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) [1,1,1,1]=>[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) [1,1,1,2]=>[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) [1,1,2,1]=>[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) [1,2,1,1]=>[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) [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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,1,1,3]=>[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) [1,1,3,1]=>[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) [1,3,1,1]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [3,1,1,1]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) [1,1,2,2]=>[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) [1,2,1,2]=>[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) [1,2,2,1]=>[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) [2,1,1,2]=>[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) [2,1,2,1]=>[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) [2,2,1,1]=>[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)=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [1,1,1,1,2]=>[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) [1,1,1,2,1]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [1,1,2,1,1]=>[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)=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) [1,2,1,1,1]=>[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)=>([(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]=>[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) [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)=>([(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
to composition
Description
Return the parking function interpreted as 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
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].