A320887 -id:A320887 - cp's OEIS frontend

A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.

1, 1, 3, 5, 10, 14, 25, 33, 54, 73, 107, 140, 207, 264, 369, 479, 652, 828, 1112, 1400, 1848, 2326, 3009, 3762, 4856, 6020, 7648, 9478, 11942, 14705, 18427, 22576, 28083, 34350, 42429, 51714, 63680, 77289, 94618, 114648, 139773, 168799, 205144, 247128, 299310, 359958, 434443, 521255, 627812, 751665, 902862

Offset: 0

Gus Wiseman, Oct 23 2018

nonn

			The a(1) = 1 through a(6) = 25 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (12)       (13)          (14)             (15)
       (1)(1)  (111)      (22)          (23)             (24)
               (1)(11)    (112)         (113)            (33)
               (1)(1)(1)  (1111)        (122)            (114)
                          (2)(2)        (1112)           (123)
                          (1)(111)      (11111)          (222)
                          (11)(11)      (2)(12)          (1113)
                          (1)(1)(11)    (1)(1111)        (1122)
                          (1)(1)(1)(1)  (11)(111)        (3)(3)
                                        (1)(1)(111)      (11112)
                                        (1)(11)(11)      (111111)
                                        (1)(1)(1)(11)    (12)(12)
                                        (1)(1)(1)(1)(1)  (2)(112)
                                                         (2)(2)(2)
                                                         (1)(11111)
                                                         (11)(1111)
                                                         (111)(111)
                                                         (1)(1)(1111)
                                                         (1)(11)(111)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(111)
                                                         (1)(1)(11)(11)
                                                         (1)(1)(1)(1)(11)
                                                         (1)(1)(1)(1)(1)(1)

Cf. A001055, A001970, A045778, A050336, A279375, A294617, A294786, A294787, A294788, A320887, A320888, A320889.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Times@@@#&]],{n,8}]

G(n)={my(M=Map()); for(k=1, n, forpart(p=k, my(t=vecprod(Vec(p)), z); mapput(M, t, if(mapisdefined(M, t, &z), z, 0) + x^k))); M}
a(n)=if(n==0, 1, vecsum(apply(p->EulerT(Vecrev(p/x, n))[n], Mat(G(n))[,2]))) \\ Andrew Howroyd, Oct 26 2018

a(13)-a(50) from Andrew Howroyd, Oct 26 2018

A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 5, 2, 2, 3, 3, 1, 5, 1, 4, 2, 2, 2, 8, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 2, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 9, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 4, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(144) = 20 set multipartitions:
  (2*3*4*6)    (2*8*9)     (2*72)     (144)
  (2*6)*(2*6)  (3*6*8)     (3*48)
  (2*6)*(3*4)  (2*3*24)    (4*36)
  (3*4)*(3*4)  (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (12)*(2*6)  (12)*(12)
               (12)*(3*4)

Cf. A001055, A001970, A045778, A050336, A052409, A089259, A294786, A296132, A319269, A320886, A320887, A320889.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[strfacs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]

a(n) = Sum_{d|A052409(n)} binomial(A045778(n^(1/d)) + d - 1, d).

A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 6, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 5, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

			The a(144) = 17 set partitions:
  (2*3*4*6)    (2*8*9)     (2*72)  (144)
  (2*6)*(3*4)  (3*6*8)     (3*48)
               (2*3*24)    (4*36)
               (2*4*18)    (6*24)
               (2*6*12)    (8*18)
               (3*4*12)    (9*16)
               (2*6)*(12)
               (3*4)*(12)

Cf. A000110, A001055, A001970, A045778, A050336, A279375, A294786, A294787, A296132, A320886, A320887, A320888.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Join@@Table[Select[sps[fac],SameQ@@Times@@@#&],{fac,strfacs[n]}]],{n,100}]

cp's OEIS Frontend

A320886 Number of multiset partitions of integer partitions of n where all parts have the same product.

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A320888 Number of set multipartitions (multisets of sets) of factorizations of n into factors > 1 such that all the parts have the same product.

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Author

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Examples

Crossrefs

Programs

Mathematica

Formula

A320889 Number of set partitions of strict factorizations of n into factors > 1 such that all the blocks have the same product.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica