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.

0, 0, 1, 0, 2, 1, 4, 3, 7, 6, 14, 14, 23, 27, 41, 47, 70, 84, 114, 141, 190, 225, 303, 370, 475, 578, 738, 890, 1131, 1368, 1698, 2058, 2549, 3048, 3759, 4505, 5495, 6574, 7966, 9483, 11450, 13606, 16307, 19351, 23116, 27297, 32470, 38293, 45346, 53342, 62939

Offset: 0

Author

Gus Wiseman, Dec 10 2020

Keywords

• nonn

Comments

The multiplicities of such a partition form a non-graphical partition.

Examples

			The a(2) = 1 through a(10) = 14 partitions (empty column indicated by dot):
  11   .   22     2111   33       2221     44         3222       55
           1111          2211     4111     2222       6111       3322
                         3111     211111   3311       222111     3331
                         111111            5111       321111     4222
                                           221111     411111     4411
                                           311111     21111111   7111
                                           11111111              222211
                                                                 322111
                                                                 331111
                                                                 421111
                                                                 511111
                                                                 22111111
                                                                 31111111
                                                                 1111111111
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:
  {{1,2},{1,2},{1,3},{1,4},{3,4}}
  {{1,2},{1,3},{1,3},{1,4},{2,4}}
  {{1,2},{1,3},{1,4},{1,4},{2,3}}
None of these are strict, so y is counted under a(22).
		

Links

• Eric Weisstein's World of Mathematics, Graphical partition.

Crossrefs

A320894 ranks these partitions (using Heinz numbers).

A338915 allows equal pairs (x,x).

A339560 counts the complement in even-length partitions.

A339564 counts factorizations of the same type.

A000070 counts non-multigraphical partitions of 2n, ranked by A339620.

A000569 counts graphical partitions, ranked by A320922.

A001358 lists semiprimes, with squarefree case A006881.

A002100 counts partitions into squarefree semiprimes.

A058696 counts partitions of even numbers, ranked by A300061.

A209816 counts multigraphical partitions, ranked by A320924.

A320655 counts factorizations into semiprimes.

A320656 counts factorizations into squarefree semiprimes.

A339617 counts non-graphical partitions of 2n, ranked by A339618.

A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.

The following count partitions of even length and give their Heinz numbers:

- A027187 has no additional conditions (A028260).

- A096373 cannot be partitioned into strict pairs (A320891).

- A338914 can be partitioned into strict pairs (A320911).

- A338915 cannot be partitioned into distinct pairs (A320892).

- A338916 can be partitioned into distinct pairs (A320912).

- A339560 can be partitioned into distinct strict pairs (A339561).

Cf. A001055, A001221, A005117, A007717, A025065, A030229, A089259, A292432, A320893, A338899, A338903, A339619.

Programs

Mathematica

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]&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&strs[Times@@Prime/@#]=={}&]],{n,0,15}]

Formula

A027187(n) = a(n) + A339560(n).

Extensions

More terms from Jinyuan Wang, Feb 14 2025