A382523 -id:A382523 - cp's OEIS frontend

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Gus Wiseman, Mar 29 2025

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

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

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[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 47

Offset: 0

Gus Wiseman, Mar 31 2025

First differs from A382523 at a(12) = 47, A382523(12) = 45.

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.

			Differs from A382523 in counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5} with partition {{1},{1,2},{1,3},{1,4},{1,5},{1,2,3}}
  {1,1,1,1,2,2,2,2,3,3,3,3} with partition {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292444.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
Weak version is A382214, complement A292432, distinct sums A382216, complement A382202.
For distinct sums we have A382523, complement A382430.
Normal multiset partitions: A034691, A035310, A116540, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A255903, A317532, A326519, A381992, A382428, A382458, A382459.

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@@#&&And@@UnsameQ@@@#&]!={}&]], {n,0,5}]

A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 32

Offset: 0

Gus Wiseman, Apr 01 2025

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.

			The a(2) = 1 through a(7) = 6 multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}  {1,1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}  {1,1,1,1,1,1,2}
                             {1,1,1,2,2}  {1,1,1,1,2,2}  {1,1,1,1,1,2,2}
                                          {1,1,1,1,2,3}  {1,1,1,1,1,2,3}
                                          {1,1,1,2,2,2}  {1,1,1,1,2,2,2}
                                                         {1,1,1,1,2,2,3}

Twice-partitions of this type are counted by A279785, strict A358914.
The strict version is A292444.
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 A381990, ranks A381806, complement A381992, ranks A382075.
The strict version for integer partitions is A382078, ranks A293243, complement A382077, ranks A382200.
The normal version is A382202, complement A382216, strict A292432, complement A382214.
The complement is counted by A382523, strict A381996.

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}]

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

Original entry on oeis.org

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}]

cp's OEIS Frontend

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

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A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

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A382430 Number of non-isomorphic finite multisets of size n that cannot be partitioned into sets with distinct sums.

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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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