OFFSET
0,4
COMMENTS
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 equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.
LINKS
Eric Weisstein's World of Mathematics, Graphical partition.
EXAMPLE
The a(2) = 1 through a(5) = 14 partitions (A = 10):
(4) (6) (8) (A)
(4,2) (4,4) (5,5)
(5,1) (5,3) (6,4)
(6,2) (7,3)
(7,1) (8,2)
(5,2,1) (9,1)
(6,1,1) (5,3,2)
(5,4,1)
(6,2,2)
(6,3,1)
(7,2,1)
(8,1,1)
(6,2,1,1)
(7,1,1,1)
For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
{{1,1},{1,1},{1,2},{2,2},{3,3}}
{{1,1},{1,1},{1,2},{2,3},{2,3}}
{{1,1},{1,1},{1,3},{2,2},{2,3}}
{{1,1},{1,2},{1,2},{1,2},{3,3}}
{{1,1},{1,2},{1,2},{1,3},{2,3}}
{{1,1},{1,2},{1,3},{1,3},{2,2}}
{{1,2},{1,2},{1,2},{1,3},{1,3}},
but since none of these is a loop-graph (because they are not strict), y is counted under a(5).
MATHEMATICA
spsbin[{}]:={{}}; spsbin[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsbin[Complement[set, s]]]/@Cases[Subsets[set], {i, _}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n], Select[mpsbin[#], UnsameQ@@#&]=={}&]], {n, 0, 5}]
CROSSREFS
A062740 counts labeled connected loop-graphs.
A101048 counts partitions into semiprimes.
A320461 ranks normal loop-graphs.
A322661 counts covering loop-graphs.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
The following count partitions of even length and give their Heinz numbers:
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 14 2020
EXTENSIONS
a(7)-a(25) from Andrew Howroyd, Jan 10 2024
STATUS
approved