Identifier
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Images
[1] => [1] => ([],1) => ([],1)
[1,1] => [1,1] => ([(0,1)],2) => ([(0,1)],2)
[2] => [2] => ([],2) => ([],1)
[1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3)
[1,2] => [1,2] => ([(1,2)],3) => ([(0,1)],2)
[2,1] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1)],2)
[3] => [3] => ([],3) => ([],1)
[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)
[1,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3)
[1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3)
[1,3] => [1,3] => ([(2,3)],4) => ([(0,1)],2)
[2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3)
[2,2] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1)],2)
[3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2)
[4] => [4] => ([],4) => ([],1)
[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)
[1,1,1,2] => [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] => [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)
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
[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)
[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] => [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)
[1,4] => [1,4] => ([(3,4)],5) => ([(0,1)],2)
[2,1,1,1] => [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)
[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)
[2,3] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1)],2)
[3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3)
[3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2)
[5] => [5] => ([],5) => ([],1)
[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)
[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] => [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)
[1,1,1,3] => [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)
[1,1,2,1,1] => [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) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
[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)
[1,1,3,1] => [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)
[1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
[1,2,1,1,1] => [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)
[1,2,1,2] => [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)
[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) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
[1,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
[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)
[1,3,2] => [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)
[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) => ([(0,1),(0,2),(1,2)],3)
[1,5] => [1,5] => ([(4,5)],6) => ([(0,1)],2)
[2,1,1,1,1] => [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)
[2,1,1,2] => [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)
[2,1,2,1] => [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)
[2,1,3] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
[2,2,1,1] => [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)
[2,2,2] => [2,2,2] => ([(1,5),(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)
[2,4] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1)],2)
[3,1,1,1] => [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)
[3,1,2] => [1,3,2] => ([(1,5),(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)
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3)
[4,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2)
[6] => [6] => ([],6) => ([],1)
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.
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].