cp's OEIS Frontend

A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

%I A339560 #19 Feb 16 2025 08:34:01
%S A339560 1,0,0,1,1,2,2,4,5,8,8,13,17,22,28,39,48,62,81,101,127,167,202,253,
%T A339560 318,395,486,608,736,906,1113,1353,1637,2011,2409,2922,3510,4227,5060,
%U A339560 6089,7242,8661,10306,12251,14503,17236,20345,24045,28334,33374,39223,46076
%N A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.
%C A339560 Naturally, such a partition must have an even number of parts. Its multiplicities form a graphical partition (A000569, A320922), and vice versa.
%H A339560 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition</a>.
%F A339560 A027187(n) = a(n) + A339559(n).
%e A339560 The a(3) = 1 through a(11) = 13 partitions (A = 10):
%e A339560   (21)  (31)  (32)  (42)  (43)    (53)    (54)    (64)    (65)
%e A339560               (41)  (51)  (52)    (62)    (63)    (73)    (74)
%e A339560                           (61)    (71)    (72)    (82)    (83)
%e A339560                           (3211)  (3221)  (81)    (91)    (92)
%e A339560                                   (4211)  (3321)  (4321)  (A1)
%e A339560                                           (4221)  (5221)  (4322)
%e A339560                                           (4311)  (5311)  (4331)
%e A339560                                           (5211)  (6211)  (4421)
%e A339560                                                           (5321)
%e A339560                                                           (5411)
%e A339560                                                           (6221)
%e A339560                                                           (6311)
%e A339560                                                           (7211)
%e A339560 For example, the partition y = (4,3,3,2,1,1) can be partitioned into a set of edges in two ways:
%e A339560   {{1,2},{1,3},{3,4}}
%e A339560   {{1,3},{1,4},{2,3}},
%e A339560 so y is counted under a(14).
%t A339560 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 A339560 Table[Length[Select[IntegerPartitions[n],strs[Times@@Prime/@#]!={}&]],{n,0,15}]
%Y A339560 A338916 allows equal pairs (x,x).
%Y A339560 A339559 counts the complement in even-length partitions.
%Y A339560 A339561 gives the Heinz numbers of these partitions.
%Y A339560 A339619 counts factorizations of the same type.
%Y A339560 A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
%Y A339560 A000569 counts graphical partitions, ranked by A320922.
%Y A339560 A001358 lists semiprimes, with squarefree case A006881.
%Y A339560 A002100 counts partitions into squarefree semiprimes.
%Y A339560 A058696 counts partitions of even numbers, ranked by A300061.
%Y A339560 A209816 counts multigraphical partitions, ranked by A320924.
%Y A339560 A320655 counts factorizations into semiprimes.
%Y A339560 A320656 counts factorizations into squarefree semiprimes.
%Y A339560 A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A339560 A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
%Y A339560 A339656 counts loop-graphical partitions, ranked by A339658.
%Y A339560 A339659 counts graphical partitions of 2n into k parts.
%Y A339560 The following count partitions of even length and give their Heinz numbers:
%Y A339560 - A027187 has no additional conditions (A028260).
%Y A339560 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339560 - A338914 can be partitioned into strict pairs (A320911).
%Y A339560 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339560 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339560 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339560 Cf. A001055, A001221, A005117, A007717, A025065, A030229, A320893, A338899, A338903, A339564.
%K A339560 nonn
%O A339560 0,6
%A A339560 _Gus Wiseman_, Dec 10 2020
%E A339560 More terms from _Jinyuan Wang_, Feb 14 2025