%I #28 Jan 11 2024 10:58:31

%S 1,2,4,8,15,28,49,84,140,229,367,577,895,1368,2064,3080,4547,6642,

%T 9627,13825,19704,27868,39164,54656,75832,104584

%N Number of 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 two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.

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

%C (1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;

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

%C (3) the unordered 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) + A339655(n).

%e The a(0) = 1 through a(4) = 15 partitions:

%e () (2) (2,2) (3,3) (3,3,2)

%e (1,1) (3,1) (2,2,2) (4,2,2)

%e (2,1,1) (3,2,1) (4,3,1)

%e (1,1,1,1) (4,1,1) (2,2,2,2)

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

%e (3,1,1,1) (3,3,1,1)

%e (2,1,1,1,1) (4,2,1,1)

%e (1,1,1,1,1,1) (5,1,1,1)

%e (2,2,2,1,1)

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

%e (4,1,1,1,1)

%e (2,2,1,1,1,1)

%e (3,1,1,1,1,1)

%e (2,1,1,1,1,1,1)

%e (1,1,1,1,1,1,1,1)

%e For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely

%e {{1,1},{2,2},{3,4}}

%e {{1,1},{2,3},{2,4}}

%e {{1,2},{1,3},{2,4}}

%e {{1,2},{1,4},{2,3}}

%e {{1,3},{1,4},{2,2}},

%e so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely

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

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

%e but neither of these is a loop-graph, so z is not counted under a(3).

%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 A339658 ranks these partitions.

%Y A001358 lists semiprimes, with squarefree case A006881.

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

%Y A027187 counts partitions of even length, ranked by A028260.

%Y A062740 counts labeled connected loop-graphs.

%Y A320461 ranks normal loop-graphs.

%Y A320655 counts factorizations into semiprimes.

%Y A322353 counts factorizations into distinct semiprimes.

%Y A322661 counts covering loop-graphs.

%Y A339845 counts the same partitions by length, or A339844 with zeros.

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

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

%Y - A000569 counts graphical partitions (A320922).

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

%Y - A209816 counts multigraphical partitions (A320924).

%Y - A321728 is conjectured to count non-half-loop-graphical partitions of n.

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

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

%Y - A339656 [this sequence] counts loop-graphical partitions (A339658).

%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. A001055, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Dec 14 2020

%E a(8)-a(25) from _Andrew Howroyd_, Jan 10 2024