%I #12 Dec 18 2020 07:58:49
%S 0,0,1,0,2,1,4,3,7,6,14,14,23,27,41,47,70,84,114,141,190,225,303,370,
%T 475,578,738,890,1131,1368,1698,2058,2549,3048,3759,4505,5495,6574,
%U 7966,9483,11450
%N Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of distinct parts, i.e., that are not the multiset union of any set of edges.
%C The multiplicities of such a partition form a non-graphical partition.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%e The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
%e 11 . 22 2111 33 2221 44 3222 55
%e 1111 2211 4111 2222 6111 3322
%e 3111 211111 3311 222111 3331
%e 111111 5111 321111 4222
%e 221111 411111 4411
%e 311111 21111111 7111
%e 11111111 222211
%e 322111
%e 331111
%e 421111
%e 511111
%e 22111111
%e 31111111
%e 1111111111
%e 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:
%e {{1,2},{1,2},{1,3},{1,4},{3,4}}
%e {{1,2},{1,3},{1,3},{1,4},{2,4}}
%e {{1,2},{1,3},{1,4},{1,4},{2,3}}
%e None of these are strict, so y is counted under a(22).
%t 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]&]}]];
%t Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]],{n,0,15}]
%Y A320894 ranks these partitions (using Heinz numbers).
%Y A338915 allows equal pairs (x,x).
%Y A339560 counts the complement in even-length partitions.
%Y A339564 counts factorizations of the same type.
%Y A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
%Y A000569 counts graphical partitions, ranked by A320922.
%Y A001358 lists semiprimes, with squarefree case A006881.
%Y A002100 counts partitions into squarefree semiprimes.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A209816 counts multigraphical partitions, ranked by A320924.
%Y A320655 counts factorizations into semiprimes.
%Y A320656 counts factorizations into squarefree semiprimes.
%Y A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
%Y The following count partitions of even length and give their Heinz numbers:
%Y - A027187 has no additional conditions (A028260).
%Y - A096373 cannot be partitioned into strict pairs (A320891).
%Y - A338914 can be partitioned into strict pairs (A320911).
%Y - A338915 cannot be partitioned into distinct pairs (A320892).
%Y - A338916 can be partitioned into distinct pairs (A320912).
%Y - A339560 can be partitioned into distinct strict pairs (A339561).
%Y Cf. A001055, A001221, A005117, A007717, A025065, A030229, A089259, A292432, A320893, A338899, A338903, A339619.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Dec 10 2020