cp's OEIS Frontend

A339559 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.

%I A339559 #19 Feb 16 2025 08:34:01
%S A339559 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 A339559 475,578,738,890,1131,1368,1698,2058,2549,3048,3759,4505,5495,6574,
%U A339559 7966,9483,11450,13606,16307,19351,23116,27297,32470,38293,45346,53342,62939
%N A339559 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 A339559 The multiplicities of such a partition form a non-graphical partition.
%H A339559 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition</a>.
%F A339559 A027187(n) = a(n) + A339560(n).
%e A339559 The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
%e A339559   11   .   22     2111   33       2221     44         3222       55
%e A339559            1111          2211     4111     2222       6111       3322
%e A339559                          3111     211111   3311       222111     3331
%e A339559                          111111            5111       321111     4222
%e A339559                                            221111     411111     4411
%e A339559                                            311111     21111111   7111
%e A339559                                            11111111              222211
%e A339559                                                                  322111
%e A339559                                                                  331111
%e A339559                                                                  421111
%e A339559                                                                  511111
%e A339559                                                                  22111111
%e A339559                                                                  31111111
%e A339559                                                                  1111111111
%e A339559 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 A339559   {{1,2},{1,2},{1,3},{1,4},{3,4}}
%e A339559   {{1,2},{1,3},{1,3},{1,4},{2,4}}
%e A339559   {{1,2},{1,3},{1,4},{1,4},{2,3}}
%e A339559 None of these are strict, so y is counted under a(22).
%t A339559 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 A339559 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]],{n,0,15}]
%Y A339559 A320894 ranks these partitions (using Heinz numbers).
%Y A339559 A338915 allows equal pairs (x,x).
%Y A339559 A339560 counts the complement in even-length partitions.
%Y A339559 A339564 counts factorizations of the same type.
%Y A339559 A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
%Y A339559 A000569 counts graphical partitions, ranked by A320922.
%Y A339559 A001358 lists semiprimes, with squarefree case A006881.
%Y A339559 A002100 counts partitions into squarefree semiprimes.
%Y A339559 A058696 counts partitions of even numbers, ranked by A300061.
%Y A339559 A209816 counts multigraphical partitions, ranked by A320924.
%Y A339559 A320655 counts factorizations into semiprimes.
%Y A339559 A320656 counts factorizations into squarefree semiprimes.
%Y A339559 A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A339559 A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
%Y A339559 The following count partitions of even length and give their Heinz numbers:
%Y A339559 - A027187 has no additional conditions (A028260).
%Y A339559 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339559 - A338914 can be partitioned into strict pairs (A320911).
%Y A339559 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339559 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339559 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339559 Cf. A001055, A001221, A005117, A007717, A025065, A030229, A089259, A292432, A320893, A338899, A338903, A339619.
%K A339559 nonn
%O A339559 0,5
%A A339559 _Gus Wiseman_, Dec 10 2020
%E A339559 More terms from _Jinyuan Wang_, Feb 14 2025