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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 distinct parts, i.e., that are not the multiset union of any set of edges.
19

%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