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Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.
23

%I #19 Feb 12 2021 11:36:20

%S 0,0,0,0,1,0,1,1,4,2,6,6,12,12,20,22,38,42,60,73,101,124,164,203,266,

%T 319,415,507,649,786,983,1198,1499,1797,2234,2673,3303,3952,4826,5753,

%U 6999

%N Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.

%C The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).

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

%F A027187(n) = a(n) + A338916(n).

%e The a(7) = 1 through a(12) = 12 partitions:

%e 211111 2222 411111 222211 222221 3333

%e 221111 21111111 331111 611111 222222

%e 311111 511111 22211111 441111

%e 11111111 22111111 32111111 711111

%e 31111111 41111111 22221111

%e 1111111111 2111111111 32211111

%e 33111111

%e 42111111

%e 51111111

%e 2211111111

%e 3111111111

%e 111111111111

%e For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:

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

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

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

%e None of these is strict, so y is counted under a(12).

%t smcs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[smcs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];

%t Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]],{n,0,10}]

%Y The Heinz numbers of these partitions are A320892.

%Y The complement in even-length partitions is A338916.

%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 A058696 counts partitions of even numbers, ranked by A300061.

%Y A209816 counts multigraphical partitions, ranked by A320924.

%Y A320655 counts factorizations into semiprimes.

%Y A322353 counts factorizations into distinct 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 A339656 counts loop-graphical partitions, ranked by 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 - 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, A007717, A025065, A320656, A320732, A320893, A338898, A338902.

%K nonn,more

%O 0,9

%A _Gus Wiseman_, Dec 10 2020