A293243 -id:A293243 - cp's OEIS frontend

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

1, 1, 1, 3, 5, 11, 23, 48, 101, 210, 436, 894

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A382216 at a(9) = 210, A382216(9) = 208.

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 {1,1,1,1,2,2,3,3,3} has partition {{1},{3},{1,2},{1,3},{1,2,3}}, so is counted under a(9).
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}

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 A292432.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
The strong version is A381996, complement A292444.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
For distinct sums we have A382216, complement A382202.
The case of a unique choice is counted by A382458, distinct sums A382459.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A317532, A326517, A326519, A381990, A381992, A382428, 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],Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A050345 Number of ways to factor n into distinct factors with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025
Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.
a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).
From _Gus Wiseman_, Apr 26 2025: (Start)
This is the number of ways to partition a factorization of n (counted by A001055) into a set of sets. For example, the a(12) = 6 choices are:
  {{2},{2,3}}
  {{2},{6}}
  {{3},{4}}
  {{2,6}}
  {{3,4}}
  {{12}}
(End)

R. J. Mathar, Table of n, a(n) for n = 1..2519

Cf. A050346-A050350. a(A002110)=A000258.
For multisets of multisets we have A050336.
For integer partitions we have a(p^k) = A050342(k), see A001970, A089259, A261049.
For normal multiset partitions see A116539, A292432, A292444, A381996, A382214, A382216.
The case of a unique choice (positions of 1) is A166684.
Twice-partitions of this type are counted by A358914, see A270995, A281113, A294788.
For sets of multisets we have A383310 (distinct products A296118).
For multisets of sets we have we have A383311, see A296119.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A302494 gives MM-numbers of sets of sets.
A382077 counts partitions that can be partitioned into a sets of sets, ranks A382200.
A382078 counts partitions that cannot be partitioned into a sets of sets, ranks A293243.
Cf. A000009, A002865, A005117, A045782, A279785, A293511, A296120, A316439, A381441, 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@@#&&And@@UnsameQ@@@#&]], {y,facs[n]}],{n,30}] (* Gus Wiseman, Apr 26 2025 *)

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).
a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017
a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 9, 13, 14, 21, 20, 32, 31, 42, 47, 63, 62, 90, 94, 117, 138, 170, 186, 235, 260, 315, 363, 429, 493, 588, 674, 795, 901, 1060, 1209, 1431, 1608, 1896, 2152, 2515, 2854, 3310, 3734, 4368, 4905, 5686

Offset: 0

Gus Wiseman, Mar 20 2025

			The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.
The a(1) = 1 through a(12) = 13 partitions:
  1  2  3  4    5    6     7    8      9      A      B      C
           211  221  411   322  332    441    433    443    552
                311  2211  331  422    522    442    533    633
                           511  611    711    622    551    822
                                3311   42111  811    722    A11
                                32111         3322   911    4422
                                              4411   42221  5511
                                              32221  53111  33321
                                              43111  62111  52221
                                              52111         54111
                                                            63111
                                                            72111
                                                            3222111

Normal multiset partitions of this type are counted by A116539, see A381718.
These partitions are ranked by A293511.
MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.
Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.
For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).
For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).
For distinct block-sums instead of blocks we have A382460, ranked by A381870.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets, see A381633.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.

ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[ssfacs[Times@@Prime/@#]]==1&]],{n,0,15}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

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}]

A339740 Non-products of distinct primes or squarefree semiprimes.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

Differs from A293243 and A212164 in having 1080, with prime indices {1,1,1,2,2,2,3} and factorization into distinct squarefree numbers 2*3*6*30.

			The sequence of terms together with their prime indices begins:
      4: {1,1}             80: {1,1,1,1,3}
      8: {1,1,1}           81: {2,2,2,2}
      9: {2,2}             88: {1,1,1,5}
     16: {1,1,1,1}         96: {1,1,1,1,1,2}
     24: {1,1,1,2}        104: {1,1,1,6}
     25: {3,3}            108: {1,1,2,2,2}
     27: {2,2,2}          112: {1,1,1,1,4}
     32: {1,1,1,1,1}      121: {5,5}
     40: {1,1,1,3}        125: {3,3,3}
     48: {1,1,1,1,2}      128: {1,1,1,1,1,1,1}
     49: {4,4}            135: {2,2,2,3}
     54: {1,2,2,2}        136: {1,1,1,7}
     56: {1,1,1,4}        144: {1,1,1,1,2,2}
     64: {1,1,1,1,1,1}    152: {1,1,1,8}
     72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
For example, a complete list of strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72); but since none of these consists of only primes or squarefree semiprimes, 72 is in the sequence.

A013929 allows only primes.
A320894 does not allow primes (but omega is assumed even).
A339741 is the complement.
A339742 has zeros at these positions.
A339840 allows squares of primes.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339661 into distinct squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339617 counts non-graphical partitions of 2n (A339618).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 counts all of even length (A028260).
- A096373/A339737 cannot be partitioned into strict pairs (A320891).
- A338915/A339662 cannot be partitioned into distinct pairs (A320892).
- A339559/A339564 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001221, A005117, A050320, A320893, A320911, A320912, A320922, A320924, A339113, A339561.

sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqps[#]=={}&]

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

Differs from A059404, A323055, A376250 in lacking 150.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.
Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

			The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.

Twice-partitions of this type (sets with a common sum) are counted by A279788.
These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
For distinct block-sums we have A381806, counted by A381990 (complement A381992).
For constant blocks we have A381871 (zeros of A381995), counted by A381993.
Partitions of this type are counted by A381994.
These are the zeros of A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
The complement counted by A383308.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A381633 counts set systems with distinct sums, see A381634, A293243.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]];
Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

Offset: 0

Gus Wiseman, Mar 29 2025

			The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
The a(1) = 1 through a(10) = 10 partitions (A=10):
  1  2  3  4    5    6     7    8      9      A
           211  221  411   322  332    441    433
                311  2211  331  422    522    442
                           511  611    711    622
                                3311   42111  811
                                32111         3322
                                              4411
                                              32221
                                              43111
                                              52111

Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633.
Normal multiset partitions of this type are counted by A381718.
These partitions are ranked by A381870.
For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
For distinct blocks instead of block-sums we have A382079, ranks A293511.
MM-numbers of these multiset partitions are A382201, see A302478.
For constant instead of strict blocks we have A382301, ranks A381991.
Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]

A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 17, 27, 43, 46, 82, 103, 133, 181, 258, 295

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			For y = (3,3,1,1) we have {{1,3},{1,3}}, so y is not counted under a(8).
For y = (3,2,2,1), although we have {{1,3},{2,2}}, the block {2,2} is not a set, so y is counted under a(8).
The a(4) = 1 through a(8) = 12 partitions:
  (2,1,1)  (2,2,1)    (4,1,1)      (3,2,2)        (3,3,2)
           (3,1,1)    (3,1,1,1)    (3,3,1)        (4,2,2)
           (2,1,1,1)  (2,1,1,1,1)  (5,1,1)        (6,1,1)
                                   (2,2,2,1)      (3,2,2,1)
                                   (3,2,1,1)      (4,2,1,1)
                                   (4,1,1,1)      (5,1,1,1)
                                   (2,2,1,1,1)    (2,2,2,1,1)
                                   (3,1,1,1,1)    (3,2,1,1,1)
                                   (2,1,1,1,1,1)  (4,1,1,1,1)
                                                  (2,2,1,1,1,1)
                                                  (3,1,1,1,1,1)
                                                  (2,1,1,1,1,1,1)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279788.
Interchanging "constant" with "strict" gives A381717, see A381635, A381636, A381991.
Normal multiset partitions of this type are counted by A381718, see A279785.
These partitions are ranked by A381719, zeros of A382080.
For distinct instead of equal block-sums we have A381990, ranked by A381806.
For constant instead of strict blocks we have A381993.
A000041 counts integer partitions, strict A000009.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A381633 counts set systems with distinct sums, see A381634, A293243.
Cf. A002846, A047966, A279786, A279789, A293511, A299202, A317142, A358914, A381992.

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[IntegerPartitions[n], Length[Select[mps[#], And@@UnsameQ@@@#&&SameQ@@Total/@#&]]==0&]],{n,0,10}]

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

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 m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {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}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

allnorm[n_Integer]:=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}]

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}]

cp's OEIS Frontend

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

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A050345 Number of ways to factor n into distinct factors with one level of parentheses.

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A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

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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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A339740 Non-products of distinct primes or squarefree semiprimes.

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A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

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A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

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A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

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A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

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