OFFSET
0,3
COMMENTS
In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.
LINKS
Eric Weisstein's World of Mathematics, Degree Sequence.
FORMULA
EXAMPLE
The a(0) = 1 through a(3) = 10 sorted degree sequences:
() (2) (1,1) (1,1,2)
(1,3) (1,1,4)
(2,2) (1,2,3)
(3,3) (1,3,4)
(2,2,2)
(2,2,4)
(2,3,3)
(2,4,4)
(3,3,4)
(4,4,4)
For example, the loop-graphs
{{1,1},{2,2},{3,3},{1,2}}
{{1,1},{2,2},{3,3},{1,3}}
{{1,1},{2,2},{3,3},{2,3}}
{{1,1},{2,2},{1,3},{2,3}}
{{1,1},{3,3},{1,2},{2,3}}
{{2,2},{3,3},{1,2},{1,3}}
all have degrees y = (3,3,2), so y is counted under a(3).
MATHEMATICA
Table[Length[Union[Sort[Table[Count[Join@@#, i], {i, n}]]&/@Select[Subsets[Subsets[Range[n], {1, 2}]/.{x_Integer}:>{x, x}], Union@@#==Range[n]&]]], {n, 0, 5}]
CROSSREFS
See link for additional cross references.
Counting the same partitions by sum gives A339656.
These partitions are ranked by A339658.
The non-covering case (zeros allowed) is A339844.
A007717 counts unlabeled multiset partitions into pairs.
A101048 counts partitions into semiprimes.
A339655 counts non-loop-graphical partitions of 2n.
A339659 counts graphical partitions of 2n into k parts.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 27 2020
EXTENSIONS
a(7)-a(12) from Andrew Howroyd, Jan 10 2024
STATUS
approved