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Matching statistic: St000644
Mapping
St000644: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 0
[3]
 => 2
[2,1]
 => 2
[1,1,1]
 => 0
[4]
 => 4
[3,1]
 => 2
[2,2]
 => 3
[2,1,1]
 => 2
[1,1,1,1]
 => 0
[5]
 => 3
[4,1]
 => 8
[3,2]
 => 8
[3,1,1]
 => 6
[2,2,1]
 => 7
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 0
[6]
 => 8
[5,1]
 => 8
[4,2]
 => 32
[4,1,1]
 => 18
[3,3]
 => 4
[3,2,1]
 => 34
[3,1,1,1]
 => 20
[2,2,2]
 => 16
[2,2,1,1]
 => 14
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 0
[7]
 => 6
[6,1]
 => 30
[5,2]
 => 40
[5,1,1]
 => 60
[4,3]
 => 42
[4,2,1]
 => 184
[4,1,1,1]
 => 82
[3,3,1]
 => 80
[3,2,2]
 => 104
[3,2,1,1]
 => 246
[3,1,1,1,1]
 => 30
[2,2,2,1]
 => 84
[2,2,1,1,1]
 => 54
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 0
[8]
 => 22
[7,1]
 => 44
[6,2]
 => 238
[6,1,1]
 => 210
[5,3]
 => 66
[5,2,1]
 => 588
Description
The number of graphs with given frequency partition. The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur. For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$. There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.