OFFSET
0,5
COMMENTS
The multiplicities of such a partition form a non-graphical partition.
LINKS
Eric Weisstein's World of Mathematics, Graphical partition.
EXAMPLE
The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
11 . 22 2111 33 2221 44 3222 55
1111 2211 4111 2222 6111 3322
3111 211111 3311 222111 3331
111111 5111 321111 4222
221111 411111 4411
311111 21111111 7111
11111111 222211
322111
331111
421111
511111
22111111
31111111
1111111111
For example, the partition y = (4,4,3,3,2,2,1,1,1,1) can be partitioned into a multiset of edges in just three ways:
{{1,2},{1,2},{1,3},{1,4},{3,4}}
{{1,2},{1,3},{1,3},{1,4},{2,4}}
{{1,2},{1,3},{1,4},{1,4},{2,3}}
None of these are strict, so y is counted under a(22).
MATHEMATICA
strs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strs[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], And[SquareFreeQ[#], PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]], {n, 0, 15}]
CROSSREFS
A320894 ranks these partitions (using Heinz numbers).
A338915 allows equal pairs (x,x).
A339560 counts the complement in even-length partitions.
A339564 counts factorizations of the same type.
A002100 counts partitions into squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
The following count partitions of even length and give their Heinz numbers:
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 10 2020
STATUS
approved