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A327012
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Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.
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5
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1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1
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OFFSET
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1,4
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COMMENTS
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Differs from A322453 at 36, 72, 100, ...
The dual of a multiset system 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}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.
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(72) = 12 factorizations:
(8*9)
(3*24)
(4*18)
(2*4*9)
(3*3*8)
(3*4*6)
(2*2*18)
(2*3*12)
(2*2*2*9)
(2*2*3*6)
(2*3*3*4)
(2*2*2*3*3)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
submultQ[cap_, fat_]:=And@@Function[i, Count[fat, i]>=Count[cap, i]]/@Union[List@@cap];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[facs[n], UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#], submultQ]&]], {n, 100}]
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CROSSREFS
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Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326975.
Partitions whose dual is a (strict) antichain are 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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