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A338915
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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.
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23
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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
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OFFSET
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0,9
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COMMENTS
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The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).
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LINKS
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Table of n, a(n) for n=0..40.
Eric Weisstein's World of Mathematics, Graphical partition.
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FORMULA
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A027187(n) = a(n) + A338916(n).
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EXAMPLE
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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).
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MATHEMATICA
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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}]
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CROSSREFS
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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.
Sequence in context: A188941 A200347 A135853 * A173197 A256568 A138947
Adjacent sequences: A338912 A338913 A338914 * A338916 A338917 A338918
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KEYWORD
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nonn,more
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AUTHOR
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Gus Wiseman, Dec 10 2020
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STATUS
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approved
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