Identifier
-
Mp00317:
Integer partitions
—odd parts⟶
Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤ
Values
[1] => 1 => [1] => ([],1) => 0
[2] => 0 => [1] => ([],1) => 0
[3] => 1 => [1] => ([],1) => 0
[2,1] => 01 => [1,1] => ([(0,1)],2) => 1
[4] => 0 => [1] => ([],1) => 0
[5] => 1 => [1] => ([],1) => 0
[4,1] => 01 => [1,1] => ([(0,1)],2) => 1
[3,2] => 10 => [1,1] => ([(0,1)],2) => 1
[2,2,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[6] => 0 => [1] => ([],1) => 0
[3,2,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[7] => 1 => [1] => ([],1) => 0
[6,1] => 01 => [1,1] => ([(0,1)],2) => 1
[5,2] => 10 => [1,1] => ([(0,1)],2) => 1
[4,3] => 01 => [1,1] => ([(0,1)],2) => 1
[4,2,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[2,2,2,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[8] => 0 => [1] => ([],1) => 0
[5,2,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[3,3,2] => 110 => [2,1] => ([(0,2),(1,2)],3) => 2
[3,2,2,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[9] => 1 => [1] => ([],1) => 0
[8,1] => 01 => [1,1] => ([(0,1)],2) => 1
[7,2] => 10 => [1,1] => ([(0,1)],2) => 1
[6,3] => 01 => [1,1] => ([(0,1)],2) => 1
[6,2,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[5,4] => 10 => [1,1] => ([(0,1)],2) => 1
[4,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[4,3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[4,2,2,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[3,3,2,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[2,2,2,2,1] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[10] => 0 => [1] => ([],1) => 0
[7,2,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[5,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[5,3,2] => 110 => [2,1] => ([(0,2),(1,2)],3) => 2
[5,2,2,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[4,3,2,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[3,2,2,2,1] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[11] => 1 => [1] => ([],1) => 0
[10,1] => 01 => [1,1] => ([(0,1)],2) => 1
[9,2] => 10 => [1,1] => ([(0,1)],2) => 1
[8,3] => 01 => [1,1] => ([(0,1)],2) => 1
[8,2,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[7,4] => 10 => [1,1] => ([(0,1)],2) => 1
[6,5] => 01 => [1,1] => ([(0,1)],2) => 1
[6,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[6,3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[6,2,2,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[5,3,2,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[4,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[4,4,2,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[4,2,2,2,1] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,3,3,2] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[3,3,2,2,1] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[2,2,2,2,2,1] => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[12] => 0 => [1] => ([],1) => 0
[9,2,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[7,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[7,3,2] => 110 => [2,1] => ([(0,2),(1,2)],3) => 2
[7,2,2,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[6,3,2,1] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[5,5,2] => 110 => [2,1] => ([(0,2),(1,2)],3) => 2
[5,4,3] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[5,4,2,1] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[5,2,2,2,1] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,3,3,2] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
[4,3,2,2,1] => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,3,3,2,1] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,2,2,2,2,1] => 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
[13] => 1 => [1] => ([],1) => 0
[12,1] => 01 => [1,1] => ([(0,1)],2) => 1
[11,2] => 10 => [1,1] => ([(0,1)],2) => 1
[10,3] => 01 => [1,1] => ([(0,1)],2) => 1
[10,2,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[9,4] => 10 => [1,1] => ([(0,1)],2) => 1
[8,5] => 01 => [1,1] => ([(0,1)],2) => 1
[8,4,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[8,3,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[8,2,2,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[7,6] => 10 => [1,1] => ([(0,1)],2) => 1
[7,3,2,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[6,6,1] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[6,5,2] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[6,4,3] => 001 => [2,1] => ([(0,2),(1,2)],3) => 2
[6,4,2,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[6,2,2,2,1] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[5,5,2,1] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[5,3,3,2] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[5,3,2,2,1] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,4,4,1] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4) => 2
[4,4,3,2] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[4,4,2,2,1] => 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,3,3,2,1] => 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[4,2,2,2,2,1] => 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,3,2,2,2,1] => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,2,2,2,2,2,1] => 0000001 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 2
[14] => 0 => [1] => ([],1) => 0
[11,2,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[9,4,1] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 1
[9,3,2] => 110 => [2,1] => ([(0,2),(1,2)],3) => 2
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Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
This is the greatest distance between any pair of vertices.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word w is the integer compositions composed of the lengths of consecutive runs of the same letter in w.
The delta morphism of a finite word w is the integer compositions composed of the lengths of consecutive runs of the same letter in w.
Map
odd parts
Description
Return the binary word indicating which parts of the partition are odd.
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.
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