%I #16 Dec 18 2020 07:58:11

%S 1,0,0,1,1,2,3,4,6,9,11,16,23,29,39,53,69,90,118,150,195,249,315,398,

%T 506,629,789,982,1219,1504,1860,2277,2798,3413,4161,5051,6137,7406,

%U 8948,10765,12943,15503,18571,22153,26432,31432,37352,44268,52444,61944,73141

%N Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

%C These are also integer partitions that can be partitioned into not necessarily distinct edges (pairs of distinct parts). For example, (3,3,2,2) can be partitioned as {{2,3},{2,3}}, so is counted under a(10), but (4,2,2,2) and (4,2,1,1,1,1) cannot be partitioned into edges. The multiplicities of such a partition form a multigraphical partition (A209816, A320924).

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

%F A027187(n) = a(n) + A096373(n).

%e The a(3) = 1 through a(10) = 11 partitions:

%e (21) (31) (32) (42) (43) (53) (54) (64)

%e (41) (51) (52) (62) (63) (73)

%e (2211) (61) (71) (72) (82)

%e (3211) (3221) (81) (91)

%e (3311) (3321) (3322)

%e (4211) (4221) (4321)

%e (4311) (4411)

%e (5211) (5221)

%e (222111) (5311)

%e (6211)

%e (322111)

%t Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]],{n,0,30}]

%Y A096373 counts the complement in even-length partitions.

%Y A320911 gives the Heinz numbers of these partitions.

%Y A339560 is the strict case.

%Y A339562 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 A320656 counts factorizations into squarefree semiprimes.

%Y A320921 counts connected graphical partitions, ranked by A320923.

%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 - 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, A001221, A005117, A007717, A030229, A320655, A322353, A338899, A338903.

%K nonn

%O 0,6

%A _Gus Wiseman_, Dec 09 2020