A292432 -id:A292432 - 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

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}]

A300442 Number of binary strict trees of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 23, 46, 108, 231, 561, 1285, 3139, 7348, 18265, 43907, 109887, 267582, 675866, 1669909, 4238462, 10555192, 26955062, 67706032, 173591181, 438555624, 1129088048, 2869732770, 7410059898, 18911818801, 48986728672, 125562853003, 326011708368

Offset: 0

Gus Wiseman, Mar 05 2018

nonn

A binary strict tree of weight n > 0 is either a single node of weight n, or an ordered pair of binary strict trees with strictly decreasing weights summing to n.

			The a(5) = 6 binary strict trees: 5, (41), (32), ((31)1), ((21)2), (((21)1)1).
The a(6) = 10 binary strict trees:
  6,
  (51), (42),
  ((41)1), ((32)1), ((31)2),
  (((31)1)1), (((21)2)1), (((21)1)2),
  ((((21)1)1)1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A292432, A293511, A300352, A300440, A300443.

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..(n-1)/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

k[n_]:=k[n]=1+Sum[Times@@k/@y,{y,Select[IntegerPartitions[n],Length[#]===2&&UnsameQ@@#&]}];
Array[k,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n - j], {j, 1, (n - 1)/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

a(n) = 1 + Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

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}]

A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 0, 3, 0, 0, 0, 5, 0, 4, 0, 1, 0, 0, 0, 6, 0, 0, 4, 0, 0, 2

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(24) = 6 strict set multipartitions of {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1},{2},{1,3,4}}
  {{1},{1,2},{3,4}}
  {{2},{1,3},{1,4}}
  {{1},{2},{3},{1,4}}

Cf. A007716, A045778, A049311, A050326, A116539, A116540, A283877, A292432 A292444.

Cf. A318283, A318285, A318287, A318360, A318361, A318362, A318369, A318370.

a(n) = A318370(A181821(n)).

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}]

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}]

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

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A300442 Number of binary strict trees of weight n.

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

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A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of 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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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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