cp's OEIS Frontend

A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.

%I A338914 #17 Feb 16 2025 08:34:01
%S A338914 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 A338914 506,629,789,982,1219,1504,1860,2277,2798,3413,4161,5051,6137,7406,
%U A338914 8948,10765,12943,15503,18571,22153,26432,31432,37352,44268,52444,61944,73141
%N A338914 Number of integer partitions of n of even length whose greatest multiplicity is at most half their length.
%C A338914 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 A338914 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A338914 A027187(n) = a(n) + A096373(n).
%e A338914 The a(3) = 1 through a(10) = 11 partitions:
%e A338914   (21)  (31)  (32)  (42)    (43)    (53)    (54)      (64)
%e A338914               (41)  (51)    (52)    (62)    (63)      (73)
%e A338914                     (2211)  (61)    (71)    (72)      (82)
%e A338914                             (3211)  (3221)  (81)      (91)
%e A338914                                     (3311)  (3321)    (3322)
%e A338914                                     (4211)  (4221)    (4321)
%e A338914                                             (4311)    (4411)
%e A338914                                             (5211)    (5221)
%e A338914                                             (222111)  (5311)
%e A338914                                                       (6211)
%e A338914                                                       (322111)
%t A338914 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Max@@Length/@Split[#]<=Length[#]/2&]],{n,0,30}]
%Y A338914 A096373 counts the complement in even-length partitions.
%Y A338914 A320911 gives the Heinz numbers of these partitions.
%Y A338914 A339560 is the strict case.
%Y A338914 A339562 counts factorizations of the same type.
%Y A338914 A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
%Y A338914 A000569 counts graphical partitions, ranked by A320922.
%Y A338914 A001358 lists semiprimes, with squarefree case A006881.
%Y A338914 A002100 counts partitions into squarefree semiprimes.
%Y A338914 A058696 counts partitions of even numbers, ranked by A300061.
%Y A338914 A209816 counts multigraphical partitions, ranked by A320924.
%Y A338914 A320656 counts factorizations into squarefree semiprimes.
%Y A338914 A320921 counts connected graphical partitions, ranked by A320923.
%Y A338914 A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A338914 A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
%Y A338914 A339656 counts loop-graphical partitions, ranked by A339658.
%Y A338914 The following count partitions of even length and give their Heinz numbers:
%Y A338914 - A027187 has no additional conditions (A028260).
%Y A338914 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A338914 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A338914 - A338916 can be partitioned into distinct pairs (A320912).
%Y A338914 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A338914 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A338914 Cf. A001055, A001221, A005117, A007717, A030229, A320655, A322353, A338899, A338903.
%K A338914 nonn
%O A338914 0,6
%A A338914 _Gus Wiseman_, Dec 09 2020