A338915 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.

0, 0, 0, 0, 1, 0, 1, 1, 4, 2, 6, 6, 12, 12, 20, 22, 38, 42, 60, 73, 101, 124, 164, 203, 266, 319, 415, 507, 649, 786, 983, 1198, 1499, 1797, 2234, 2673, 3303, 3952, 4826, 5753, 6999, 8330, 10051, 11943, 14357, 16956, 20322, 23997, 28568, 33657, 39897, 46879

Offset: 0

Author

Gus Wiseman, Dec 10 2020

Keywords

• nonn

Comments

The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).

Examples

			The a(7) = 1 through a(12) = 12 partitions:
  211111  2222      411111    222211      222221      3333
          221111    21111111  331111      611111      222222
          311111              511111      22211111    441111
          11111111            22111111    32111111    711111
                              31111111    41111111    22221111
                              1111111111  2111111111  32211111
                                                      33111111
                                                      42111111
                                                      51111111
                                                      2211111111
                                                      3111111111
                                                      111111111111
For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}}
None of these is strict, so y is counted under a(12).
		

Links

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

Crossrefs

The Heinz numbers of these partitions are A320892.

The complement in even-length partitions is A338916.

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

A000569 counts graphical partitions, ranked by A320922.

A001358 lists semiprimes, with squarefree case A006881.

A058696 counts partitions of even numbers, ranked by A300061.

A209816 counts multigraphical partitions, ranked by A320924.

A320655 counts factorizations into semiprimes.

A322353 counts factorizations into distinct semiprimes.

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

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

A339656 counts loop-graphical partitions, ranked by A339658.

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).

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

- A339559 cannot be partitioned into distinct strict pairs (A320894).

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

Cf. A001055, A007717, A025065, A320656, A320732, A320893, A338898, A338902.

Programs

Mathematica

smcs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[smcs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]],{n,0,10}]

Formula

A027187(n) = a(n) + A338916(n).

Extensions

More terms from Jinyuan Wang, Feb 14 2025