cp's OEIS Frontend

A383311 Number of ways to choose a set multipartition (multiset of sets) of a factorization of n into factors > 1.

%I A383311 #5 Apr 28 2025 13:04:16
%S A383311 1,1,1,2,1,3,1,4,2,3,1,7,1,3,3,7,1,7,1,7,3,3,1,16,2,3,4,7,1,12,1,12,3,
%T A383311 3,3,20,1,3,3,16,1,12,1,7,7,3,1,33,2,7,3,7,1,16,3,16,3,3,1,34,1,3,7,
%U A383311 22,3,12,1,7,3,12,1,49,1,3,7,7,3,12,1,33,7,3
%N A383311 Number of ways to choose a set multipartition (multiset of sets) of a factorization of n into factors > 1.
%C A383311 First differs from A296119 at a(36) = 20, A296119(36) = 21.
%e A383311 The a(36) = 20 choices are:
%e A383311   {{2,3,6}}  {{2,3},{2,3}}  {{2},{3},{2,3}}  {{2},{2},{3},{3}}
%e A383311   {{2,18}}   {{2},{2,9}}    {{2},{2},{9}}
%e A383311   {{3,12}}   {{2},{3,6}}    {{2},{3},{6}}
%e A383311   {{4,9}}    {{3},{2,6}}    {{3},{3},{4}}
%e A383311   {{36}}     {{6},{2,3}}
%e A383311              {{2},{18}}
%e A383311              {{3},{3,4}}
%e A383311              {{3},{12}}
%e A383311              {{4},{9}}
%e A383311              {{6},{6}}
%t A383311 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A383311 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A383311 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A383311 Table[Sum[Length[Select[mps[y], And@@UnsameQ@@@#&]], {y,facs[n]}],{n,100}]
%Y A383311 The case of a unique choice (positions of 1) is A008578.
%Y A383311 For multisets of multisets we have A050336.
%Y A383311 For sets of sets we have A050345.
%Y A383311 For normal multisets we have A116540, strong A330783.
%Y A383311 For integer partitions instead of factorizations we have A089259.
%Y A383311 Twice-partitions of this type are counted by A270995.
%Y A383311 For sets of multisets we have A383310 (distinct products A296118).
%Y A383311 A001055 counts factorizations, strict A045778.
%Y A383311 A050320 counts factorizations into squarefree numbers, distinct A050326.
%Y A383311 A281113 counts twice-factorizations, see A294788, A296120, A296121.
%Y A383311 A302478 gives MM-numbers of set multipartitions.
%Y A383311 A302494 gives MM-numbers of sets of sets.
%Y A383311 Cf. A000009, A001970, A005117, A050342, A116539, A279785, A293243, A293511, A296119, A316439, A381992, A382077.
%K A383311 nonn
%O A383311 1,4
%A A383311 _Gus Wiseman_, Apr 28 2025