A050342 -id:A050342 - cp's OEIS frontend

A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

1, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 0, 7, 11, 1, 0, 0, 0, 11, 20, 6, 0, 0, 0, 0, 15, 40, 16, 0, 0, 0, 0, 0, 22, 68, 40, 3, 0, 0, 0, 0, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
  1
  0  1
  0  2  0
  0  3  2  0
  0  5  4  0  0
  0  7 11  1  0  0
  0 11 20  6  0  0  0
  0 15 40 16  0  0  0  0
  0 22 68 40  3  0  0  0  0
  ...
Row n = 5 counts the following sets of multisets:
  {{5}}          {{1},{4}}        {{1},{2},{1,1}}
  {{1,4}}        {{2},{3}}
  {{2,3}}        {{1},{1,3}}
  {{1,1,3}}      {{1},{2,2}}
  {{1,2,2}}      {{2},{1,2}}
  {{1,1,1,2}}    {{3},{1,1}}
  {{1,1,1,1,1}}  {{1},{1,1,2}}
                 {{1,1},{1,2}}
                 {{2},{1,1,1}}
                 {{1},{1,1,1,1}}
                 {{1,1},{1,1,1}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Row sums are A261049.
Column k = 1 is A000041.
Multisets of multisets are A061260, with row sums A001970.
Sets of sets are A330462, with row sums A050342.
Multisets of sets are A285229, with row sums A089259.
Sets of disjoint sets are A330460, with row sums A294617.
Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
       combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 30 2019

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],And[UnsameQ@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
     PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - Andrew Howroyd, Dec 29 2019

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 30, 61, 110, 207, 381, 711, 1250

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 17 multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,1,1}}    {{2,2}}        {{1,1,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,2,2}}
                  {{1},{1,1}}  {{1},{3}}      {{1},{4}}
                               {{1,1,1,1}}    {{2},{3}}
                               {{1},{1,2}}    {{1,1,1,2}}
                               {{2},{1,1}}    {{1},{1,3}}
                               {{1},{1,1,1}}  {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A319066, A319312, A320328, A320330.

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]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,UnsameQ@@dual[#]]&]],{n,8}]

A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 17 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{2},{1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1},{2,2}}
                                   {{1},{1},{2}}      {{3},{1,1}}
                                   {{1},{1},{1},{1}}  {{1,1,1,1,1}}
                                                      {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, A320355, A320356.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],antiQ]],{n,8}]

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}]

A089254 G.f.: Product_{m>=1} 1/(1+x^m)^A000009(m).

Original entry on oeis.org

1, -1, 0, -2, 1, -2, 2, -2, 5, -3, 8, -3, 13, -6, 16, -15, 21, -28, 21, -52, 22, -90, 23, -136, 40, -190, 79, -237, 185, -254, 385, -219, 740, -116, 1279, 49, 2039, 198, 2923, 130, 3785, -541, 4215, -2464, 3638, -6581, 1204, -14062, -3889, -26110, -12267, -43399, -23264, -65206, -34085, -87955, -37666

Offset: 0

N. J. A. Sloane, Dec 23 2003

sign

Cf. A050342, A000009, A001970, A089259.

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946

Offset: 0

Gus Wiseman, Oct 12 2018

nonn

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022

			The a(9) = 26 multiset partitions:
  {{9}}
  {{1,8}}
  {{2,7}}
  {{3,6}}
  {{4,5}}
  {{1,2,6}}
  {{1,3,5}}
  {{1},{8}}
  {{2,3,4}}
  {{2},{7}}
  {{3,3,3}}
  {{3},{6}}
  {{4},{5}}
  {{1},{2},{6}}
  {{1},{3},{5}}
  {{2},{3},{4}}
  {{3},{3},{3}}
  {{1,1,1,2,2,2}}
  {{1,1,1},{2,2,2}}
  {{1,1,2},{1,2,2}}
  {{1,1},{1,2},{2,2}}
  {{1,2},{1,2},{1,2}}
  {{1,1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,1,1},{1,1,1}}
  {{1},{1},{1},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..150
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.

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],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]

Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022

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}]

A330461 Array read by antidiagonals where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 4, 7, 7, 5, 1, 1, 1, 1, 5, 12, 14, 11, 6, 1, 1, 1, 1, 6, 19, 29, 25, 16, 7, 1, 1, 1, 1, 8, 30, 57, 60, 41, 22, 8, 1, 1, 1, 1, 10, 49, 110, 141, 111, 63, 29, 9, 1, 1, 1

Offset: 0

Gus Wiseman, Dec 18 2019

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6
      -----------------------------
  n=0:  1   1   1   1   1   1   1
  n=1:  1   1   1   1   1   1   1
  n=2:  1   1   1   1   1   1   1
  n=3:  1   2   3   4   5   6   7
  n=4:  1   2   4   7  11  16  22
  n=5:  1   3   7  14  25  41  63
  n=6:  1   4  12  29  60 111 189
For example, the A(5,3) = 14 partitions are:
  {{5}}      {{1}}{{4}}
  {{14}}     {{2}}{{3}}
  {{23}}     {{1}}{{13}}
  {{1}{4}}   {{2}}{{12}}
  {{2}{3}}   {{1}}{{1}{3}}
  {{1}{13}}  {{2}}{{1}{2}}
  {{2}{12}}  {{1}}{{1}{12}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns are A000012 (k = 0), A000009 (k = 1), A050342 (k = 2), A050343 (k = 3), A050344 (k = 4).
The non-strict version is A290353.
Cf. A001970, A004111, A007713, A060016, A273873, A279375, A279785, A294617, A306186, A323718, A323790, A330462.

spl[n_,0]:={n};
spl[n_,k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}],UnsameQ@@#&];
Table[Length[spl[n-k,k]],{n,0,10},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
M(n, k=n)={my(L=List(), v=vector(n,i,1)); listput(L, concat([1], v)); for(j=1, k, v=WeighT(v); listput(L, concat([1], v))); Mat(Col(L))~}
{ my(A=M(7)); for(i=1, #A, print(A[i,])) } \\ Andrew Howroyd, Dec 31 2019

Column k is the k-th weigh transform of the all-ones sequence. The weigh transform of a sequence b has generating function Product_{i > 0} (1 + x^i)^b(i).

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

A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

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A320331 Number of strict T_0 multiset partitions of integer partitions of n.

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A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

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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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A089254 G.f.: Product_{m>=1} 1/(1+x^m)^A000009(m).

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A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

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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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A330461 Array read by antidiagonals where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.

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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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