A293243 -id:A293243 - cp's OEIS frontend

A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9

Offset: 0

Gus Wiseman, Mar 30 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}

For constant instead of strict blocks we have A000045.
Factorizations of this type are counted by A050326, with distinct sums A381633.
For the strong case see A292444, A382430, complement A381996, A382523.
MM-numbers of sets of sets are A302494, see A302478, A382201.
Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.
For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).
With distinct sums we have A382459.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n], Length[Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]

A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 2, 7, 4, 10, 19

Offset: 0

Gus Wiseman, Apr 01 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,2,2,2,2,3,3,4} has only one multiset partition into a set of sets with distinct sums: {{2},{1,2},{2,3},{2,3,4}}, so is counted under a(8).
The a(1) = 1 through a(7) = 7 multisets:
  {1}  .  {112}  {1122}  {11123}  {111233}  {1111234}
          {122}          {12223}  {122233}  {1112223}
                         {12333}            {1112333}
                                            {1222234}
                                            {1222333}
                                            {1233334}
                                            {1234444}

Twice-partitions of this type are counted by A279785, A270995, A358914.
Factorizations of this type are counted by A381633, A050320, A050326.
Normal multiset partitions of this type are A381718, A116540, A116539.
Multiset partitions of this type are ranked by A382201, A302478, A302494.
For at least one choice: A382216 (strict A382214), complement A382202 (strict A292432).
For the strong case see: A382430 (strict A292444), complement A382523 (strict A381996).
Without distinct sums we have A382458.
For integer partitions we have A382460, ranks A381870, strict A382079, ranks A293511.
Set multipartitions: A089259, A296119, A318360.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A317532, A321469, A326519, A381078, A381441, A382428.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]]==1&]],{n,0,5}]

A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 45

Offset: 0

Gus Wiseman, Apr 01 2025

First differs from A381996 at a(12) = 45, A381996(12) = 47.

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

			First differs from A381996 in not counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5}
  {1,1,1,1,2,2,2,2,3,3,3,3}
The a(1) = 1 through a(6) = 6 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  {1,1,1,2,2,3}
              {1,2,3}  {1,1,2,3}  {1,1,2,2,3}  {1,1,1,2,3,4}
                       {1,2,3,4}  {1,1,2,3,4}  {1,1,2,2,3,3}
                                  {1,2,3,4,5}  {1,1,2,2,3,4}
                                               {1,1,2,3,4,5}
                                               {1,2,3,4,5,6}

Twice-partitions of this type are counted by A279785, strict A358914.
Factorizations of this type are counted by A381633, strict A050326.
Normal multiset partitions of this type are counted by A381718, strict A116539.
For integer partitions we have A381992, ranks A382075, complement A381990, ranks A381806.
The strict version is A381996.
The strict version for integer partitions is A382077, ranks A382200, complement A382078, ranks A293243.
The labeled version is A382216, complement A382202, strict A382214, complement A292432.
The complement is counted by A382430, strict A292444.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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[strnorm[n],Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A050327 Number of factorizations into distinct squarefree factors indexed by prime signatures. A050326(A025487).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 0, 5, 0, 1, 0, 4, 0, 0, 0, 1, 0, 0, 5, 0, 15, 0, 0, 0, 0, 2, 0, 16, 0, 0, 0, 0, 0, 0, 7, 0, 8, 0, 0, 1, 0, 23, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 52, 14, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 68, 3, 0, 4, 0, 0, 40, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 41

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

From Michael De Vlieger, Oct 06 2017: (Start)
Terms in A025487 that set records in this sequence: {1, 6, 30, 210, 420, 1260, 2310, 4620, 13860, 30030, 60060, 180180, 510510, 900900, 1021020, 3063060, 6126120, 9699690, ...}.
Conjecture: prime signatures corresponding to primorials A002110(i) with i > 1 set records in this sequence. (End)

			From _Michael De Vlieger_, Oct 06 2017: (Start)
First 20 values, with numbers in column "r" records, and the last column the concatenation of exponents of standard form prime decomposition of A025487(n):
  .
   n    a(n)   r   A025487(n) rev(A054841(A025487(n)))
  --------------------------------------------
   1      1    1          1   0
   2      1               2   1
   3      0               4   2
   4      2    2          6   11
   5      0               8   3
   6      1              12   21
   7      0              16   4
   8      0              24   31
   9      5    3         30   111
  10      0              32   5
  11      1              36   22
  12      0              48   41
  13      4              60   211
  14      0              64   6
  15      0              72   32
  16      0              96   51
  17      1             120   311
  18      0             128   7
  19      0             144   42
  20      5             180   221
  21      0             192   61
  22     15    4        210   1111
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..500
Michael De Vlieger, Relations between A050327, A025487, and A002110.

Cf. A000110, A002110, A025487, A050326.

f[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[f[n/d], Min @@ # > d &]], {d, Select[Rest@ Divisors@ n, SquareFreeQ]}]]; Length[f@ #] & /@ Prepend[#, 1] &@ Sort@ Map[Times @@ Flatten@ MapIndexed[ConstantArray[Prime@ First@ #2, #1] &, #] &, Union@ Table[Sort[FactorInteger[n][[All, -1]], Greater], {n, 2, Product[Prime@ i, {i, 7}]}]] (* Michael De Vlieger, Oct 06 2017, after Gus Wiseman at A293243 *)

From Michael De Vlieger, Oct 06 2017: (Start)
a(n) = A050326(A025487(n)).
A050326(A002110(n)) = A000110(n).
(End)

A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 9, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 19, 3, 3, 3, 24, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 46, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 37, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 46, 9, 3

Offset: 1

Gus Wiseman, Apr 26 2025

nonn

			The a(36) = 24 choices:
  {{2,2,3,3}}  {{2},{2,3,3}}  {{2},{3},{2,3}}
  {{2,2,9}}    {{3},{2,2,3}}  {{2},{3},{6}}
  {{2,3,6}}    {{2,2},{3,3}}
  {{2,18}}     {{2},{2,9}}
  {{3,3,4}}    {{9},{2,2}}
  {{3,12}}     {{2},{3,6}}
  {{4,9}}      {{3},{2,6}}
  {{6,6}}      {{6},{2,3}}
  {{36}}       {{2},{18}}
               {{3},{3,4}}
               {{4},{3,3}}
               {{3},{12}}
               {{4},{9}}

The case of a unique choice (positions of 1) is A008578.
This is the strict case of A050336.
For distinct strict blocks we have A050345.
For integer partitions we have A261049, strict case of A001970.
For strict blocks that are not necessarily distinct we have A296119.
Twice-partitions of this type are counted by A296122.
For normal multisets we have A317776, strict case of A255906.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, strict A296121, see A296118, A296120.
Cf. A000009, A005117, A045782, A050342, A279785, A293243, A293511, A302494, A316439, A358914, A381992, A382201.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[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,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y],UnsameQ@@#&]],{y,facs[n]}],{n,30}]

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

Original entry on oeis.org

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, 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, 22, 3, 12, 1, 7, 3, 12, 1, 49, 1, 3, 7, 7, 3, 12, 1, 33, 7, 3

Offset: 1

Gus Wiseman, Apr 28 2025

nonn

First differs from A296119 at a(36) = 20, A296119(36) = 21.

			The a(36) = 20 choices are:
  {{2,3,6}}  {{2,3},{2,3}}  {{2},{3},{2,3}}  {{2},{2},{3},{3}}
  {{2,18}}   {{2},{2,9}}    {{2},{2},{9}}
  {{3,12}}   {{2},{3,6}}    {{2},{3},{6}}
  {{4,9}}    {{3},{2,6}}    {{3},{3},{4}}
  {{36}}     {{6},{2,3}}
             {{2},{18}}
             {{3},{3,4}}
             {{3},{12}}
             {{4},{9}}
             {{6},{6}}

The case of a unique choice (positions of 1) is A008578.
For multisets of multisets we have A050336.
For sets of sets we have A050345.
For normal multisets we have A116540, strong A330783.
For integer partitions instead of factorizations we have A089259.
Twice-partitions of this type are counted by A270995.
For sets of multisets we have A383310 (distinct products A296118).
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, see A294788, A296120, A296121.
A302478 gives MM-numbers of set multipartitions.
A302494 gives MM-numbers of sets of sets.
Cf. A000009, A001970, A005117, A050342, A116539, A279785, A293243, A293511, A296119, A316439, A381992, A382077.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[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,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y], And@@UnsameQ@@@#&]], {y,facs[n]}],{n,100}]

cp's OEIS Frontend

A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

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A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

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A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.

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A050327 Number of factorizations into distinct squarefree factors indexed by prime signatures. A050326(A025487).

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A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

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

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