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A326978
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Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.
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10
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1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 52, 68, 91, 116, 149, 191, 249, 311, 399, 498, 622, 773, 971, 1193, 1478, 1811, 2222, 2709, 3311, 4021, 4882, 5894, 7110, 8554, 10273, 12312, 14734, 17578, 20941, 24905, 29570, 35056, 41475, 48983, 57752, 68025, 79988
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OFFSET
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0,3
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COMMENTS
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The dual of a multiset partition has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}.
A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.
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LINKS
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EXAMPLE
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The a(0) = 1 through a(7) = 15 partitions:
() (1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (43)
(111) (31) (41) (42) (52)
(211) (221) (51) (61)
(1111) (311) (222) (322)
(2111) (321) (331)
(11111) (411) (421)
(2211) (511)
(3111) (2221)
(21111) (3211)
(111111) (4111)
(22111)
(31111)
(211111)
(1111111)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], stableQ[dual[primeMS/@#], submultQ]&]], {n, 0, 30}]
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CROSSREFS
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Set-systems whose dual is a weak antichain are A326968.
Factorizations whose dual is a weak antichain are A326975.
The version where the dual is a strict antichain is A326977.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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