Identifier
-
Mp00184:
Integer compositions
—to threshold graph⟶
Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤ
Values
[2] => ([],2) => [1,1] => [1] => 0
[1,2] => ([(1,2)],3) => [2,1] => [1] => 0
[3] => ([],3) => [1,1,1] => [1,1] => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => [3,1] => [1] => 0
[1,3] => ([(2,3)],4) => [2,1,1] => [1,1] => 1
[2,2] => ([(1,3),(2,3)],4) => [3,1] => [1] => 0
[4] => ([],4) => [1,1,1,1] => [1,1,1] => 2
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 0
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1] => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 0
[1,4] => ([(3,4)],5) => [2,1,1,1] => [1,1,1] => 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1] => 0
[2,3] => ([(2,4),(3,4)],5) => [3,1,1] => [1,1] => 1
[3,2] => ([(1,4),(2,4),(3,4)],5) => [4,1] => [1] => 0
[5] => ([],5) => [1,1,1,1,1] => [1,1,1,1] => 3
[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) => [5,1] => [1] => 0
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => 2
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
[1,5] => ([(4,5)],6) => [2,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
[2,4] => ([(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => 2
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1] => 0
[3,3] => ([(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1] => 0
[6] => ([],6) => [1,1,1,1,1,1] => [1,1,1,1,1] => 4
[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) => [6,1] => [1] => 0
[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) => [5,1,1] => [1,1] => 1
[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) => [6,1] => [1] => 0
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[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) => [6,1] => [1] => 0
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => 3
[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) => [6,1] => [1] => 0
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[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) => [6,1] => [1] => 0
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
[1,6] => ([(5,6)],7) => [2,1,1,1,1,1] => [1,1,1,1,1] => 4
[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) => [6,1] => [1] => 0
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[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) => [6,1] => [1] => 0
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[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) => [6,1] => [1] => 0
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
[2,5] => ([(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => 3
[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) => [6,1] => [1] => 0
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
[3,4] => ([(3,6),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6,1] => [1] => 0
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6,1] => [1] => 0
[7] => ([],7) => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 5
[1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,1,3,2] => ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [4,1,1,1,1] => [1,1,1,1] => 3
[1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,2,2,2] => ([(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,1,1,2,2] => ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[2,1,3,2] => ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [7,1] => [1] => 0
[4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
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search for individual values
searching the database for the individual values of this statistic
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
This is the number of boxes in its diagram that are not in the first row.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
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.
Map
first row removal
Description
Removes the first entry of an integer partition
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