%I #13 Jan 11 2024 10:58:35

%S 0,0,1,3,7,14,28,51,91,156,260,425,680,1068,1654,2524,3802,5668,8350,

%T 12190,17634,25306,36011,50902,71441,99642

%N Number of non-loop-graphical integer partitions of 2n.

%C 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.

%C The following are equivalent characteristics for any positive integer n:

%C (1) the prime factors of n can be partitioned into distinct pairs;

%C (2) n can be factored into distinct semiprimes;

%C (3) the prime signature of n is loop-graphical.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>

%F A058696(n) = a(n) + A339656(n).

%e The a(2) = 1 through a(5) = 14 partitions (A = 10):

%e (4) (6) (8) (A)

%e (4,2) (4,4) (5,5)

%e (5,1) (5,3) (6,4)

%e (6,2) (7,3)

%e (7,1) (8,2)

%e (5,2,1) (9,1)

%e (6,1,1) (5,3,2)

%e (5,4,1)

%e (6,2,2)

%e (6,3,1)

%e (7,2,1)

%e (8,1,1)

%e (6,2,1,1)

%e (7,1,1,1)

%e For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:

%e {{1,1},{1,1},{1,2},{2,2},{3,3}}

%e {{1,1},{1,1},{1,2},{2,3},{2,3}}

%e {{1,1},{1,1},{1,3},{2,2},{2,3}}

%e {{1,1},{1,2},{1,2},{1,2},{3,3}}

%e {{1,1},{1,2},{1,2},{1,3},{2,3}}

%e {{1,1},{1,2},{1,3},{1,3},{2,2}}

%e {{1,2},{1,2},{1,2},{1,3},{1,3}},

%e but since none of these is a loop-graph (because they are not strict), y is counted under a(5).

%t spsbin[{}]:={{}};spsbin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];

%t mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];

%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];

%t Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]=={}&]],{n,0,5}]

%Y A001358 lists semiprimes, with squarefree case A006881.

%Y A006125 counts labeled graphs, with covering case A006129.

%Y A062740 counts labeled connected loop-graphs.

%Y A101048 counts partitions into semiprimes.

%Y A320461 ranks normal loop-graphs.

%Y A322661 counts covering loop-graphs.

%Y A320655 counts factorizations into semiprimes.

%Y The following count vertex-degree partitions and give their Heinz numbers:

%Y - A058696 counts partitions of 2n (A300061).

%Y - A000070 counts non-multigraphical partitions of 2n (A339620).

%Y - A209816 counts multigraphical partitions (A320924).

%Y - A339655 (this sequence) counts non-loop-graphical partitions of 2n (A339657).

%Y - A339656 counts loop-graphical partitions (A339658).

%Y - A339617 counts non-graphical partitions of 2n (A339618).

%Y - A000569 counts graphical partitions (A320922).

%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 - A339559 cannot be partitioned into distinct strict pairs (A320894).

%Y - A339560 can be partitioned into distinct strict pairs (A339561).

%Y Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Dec 14 2020

%E a(7)-a(25) from _Andrew Howroyd_, Jan 10 2024