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A326975
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Number of factorizations of n into factors > 1 whose dual is a weak antichain.
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10
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1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 11, 2, 5, 1, 2, 2, 5, 1, 12, 1, 2, 2, 2, 2, 5, 1, 5, 5, 2, 1, 4, 2, 2
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OFFSET
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1,4
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COMMENTS
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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.
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(36) = 9 factorizations:
(36)
(4*9)
(6*6)
(2*18)
(3*12)
(2*2*9)
(2*3*6)
(3*3*4)
(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], stableQ[dual[primeMS/@#], submultQ]&]], {n, 100}]
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CROSSREFS
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The T_0 case (where the dual is strict) is A316978.
Set-systems whose dual is a weak antichain are A326968.
Partitions whose dual is a weak antichain are A326978.
The T_1 case (where the dual is a strict antichain) is A327012.
Cf. A001055, A059523, A316978, A319639, A326965, A326966, A326969, A326970, A326971, A326976, 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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