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A326977
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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 (strict) antichain, also called T_1 integer partitions.
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14
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1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 49, 64, 85, 109, 141, 181, 234, 294, 375, 470, 589, 733, 917, 1131, 1401, 1720, 2113, 2581, 3153, 3833, 4655, 5631, 6801, 8192, 9849, 11816, 14148, 16899, 20153, 23990, 28503, 33815, 40038, 47330, 55858, 65841, 77475
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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}}. An antichain is a set of multisets, none of which is a 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) = 14 partitions:
() (1) (2) (3) (4) (5) (33) (7)
(11) (21) (22) (32) (42) (43)
(111) (31) (41) (51) (52)
(211) (221) (222) (322)
(1111) (311) (321) (331)
(2111) (411) (421)
(11111) (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], UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#], submultQ]&]], {n, 0, 30}]
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CROSSREFS
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T_0 integer partitions are A319564.
Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326978.
T_1 factorizations (whose dual is a strict antichain) are A327012.
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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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