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A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

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)

MATHEMATICA

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}]

CROSSREFS

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.

Cf. A001055, A316978, A326961, A326974, A326976, A326979.

Sequence in context: A000688 A295879 A322453 * A328855 A327658 A319786

Adjacent sequences:  A327009 A327010 A327011 * A327013 A327014 A327015

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 13 2019

STATUS

approved

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Last modified September 29 12:15 EDT 2020. Contains 337431 sequences. (Running on oeis4.)