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
St000937: Integer partitions ⟶ ℤ
Values
[3] => ([],3) => [1,1,1] => [1,1] => 1
[1,3] => ([(2,3)],4) => [2,1,1] => [1,1] => 1
[4] => ([],4) => [1,1,1,1] => [1,1,1] => 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,1] => 1
[1,4] => ([(3,4)],5) => [2,1,1,1] => [1,1,1] => 2
[2,3] => ([(2,4),(3,4)],5) => [3,1,1] => [1,1] => 1
[5] => ([],5) => [1,1,1,1,1] => [1,1,1,1] => 3
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[1,5] => ([(4,5)],6) => [2,1,1,1,1] => [1,1,1,1] => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[2,4] => ([(3,5),(4,5)],6) => [3,1,1,1] => [1,1,1] => 2
[3,3] => ([(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,1] => 1
[6] => ([],6) => [1,1,1,1,1,1] => [1,1,1,1,1] => 4
[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
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => 3
[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
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[1,6] => ([(5,6)],7) => [2,1,1,1,1,1] => [1,1,1,1,1] => 4
[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
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[2,5] => ([(4,6),(5,6)],7) => [3,1,1,1,1] => [1,1,1,1] => 3
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[3,4] => ([(3,6),(4,6),(5,6)],7) => [4,1,1,1] => [1,1,1] => 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => [5,1,1] => [1,1] => 1
[7] => ([],7) => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[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
[1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[1,2,2,3] => ([(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
[1,3,1,3] => ([(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
[1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[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
[2,2,1,3] => ([(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
[2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[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
[3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8) => [6,1,1] => [1,1] => 1
[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
[4,4] => ([(3,7),(4,7),(5,7),(6,7)],8) => [5,1,1,1] => [1,1,1] => 2
[8] => ([],8) => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 8
search for individual values
searching the database for the individual values of this statistic
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
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
searching the database
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