A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

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

Offset

1,4

Comments

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.

Links

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

Example

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)

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], stableQ[dual[primeMS/@#], submultQ]&]], {n, 100}]

Crossrefs

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.

Sequence in context: A328195 A235875 A328026 * A204893 A251717 A057217

Adjacent sequences: A326972 A326973 A326974 * A326976 A326977 A326978

Keyword

nonn

Author

Gus Wiseman, Aug 13 2019

Status

approved