A316980 -id:A316980 - cp's OEIS frontend

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

1, 1, 2, 4, 11, 26, 77, 220, 677, 2098, 6756, 22101, 74264, 253684, 883795, 3130432, 11275246, 41240180, 153117873, 576634463, 2201600769, 8517634249, 33378499157, 132438117118, 531873247805, 2161293783123, 8883906870289, 36928576428885, 155196725172548, 659272353608609, 2830200765183775

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be cross-balanced if it uses exactly as many distinct vertices as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{2,2}}    {{1,1},{2,2}}
                    {{2},{1,2}}    {{1,2},{1,2}}
                    {{1},{1},{1}}  {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{2,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{2,2}}
                                   {{2},{2},{1,2}}
                                   {{1},{1},{1},{1}}

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

The co-balanced version is A319616.
The balanced version is A340600.
The twice-balanced version is A340652.
The version for factorizations is A340654.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A047993 counts balanced partitions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340653 counts balanced factorizations.
Cf. A001055, A007717, A064174, A316983, A320663, A324522, A339888, A340655.

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n, G(k,n,k) - G(k-1,n,k) - G(k,n,k-1) + G(k-1,n,k-1)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A319751 Number of non-isomorphic set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 2, 6, 13, 35, 83, 217, 556, 1504, 4103, 11715, 34137, 103155, 320217, 1025757, 3376889, 11436712, 39758152, 141817521, 518322115, 1939518461, 7422543892, 29028055198, 115908161428, 472185530376, 1961087909565, 8298093611774, 35750704171225, 156734314212418

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

A set system is a finite set of finite nonempty sets. Its weight is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 set systems:
2: {{1},{2}}
3: {{1},{2,3}}
   {{1},{2},{3}}
4: {{1},{2,3,4}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}
5: {{1},{2,3,4,5}}
   {{1,2},{3,4,5}}
   {{1},{2},{3,4,5}}
   {{1},{4},{2,3,4}}
   {{1},{2,3},{4,5}}
   {{1},{2,4},{3,4}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{4},{1,2},{3,4}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{3},{4,5}}
   {{1},{2},{4},{3,4}}
   {{1},{2},{3},{4},{5}}

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

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t]-subst(u[t],x,x^2), O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t] - subst(x^t*u[t],x,x^2), O(x*x^n)))*(1+x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 3, 5, 7, 14, 25

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. 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.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 7 multiset partitions:
1: {{1}}
3: {{2},{1,2}}
4: {{1,3},{2,3}}
5: {{3},{1,3},{2,3}}
6: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,4},{2,4},{3,4}}
7: {{4},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4}}
   {{1,3},{2,3},{1,2,3}}
   {{1,4},{3,4},{2,3,4}}
   {{4},{1,4},{2,4},{3,4}}
8: {{1,5},{2,4,5},{3,4,5}}
   {{2,4},{3,4},{1,2,3,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{2,4},{1,3,4},{2,3,4}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{4},{1,4},{3,4},{2,3,4}}
   {{1,5},{2,5},{3,5},{4,5}}

Cf. A007716, A049311, A283877, A305843, A305854, A306006, A316980, A317752.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

A321402 Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 8, 14, 27, 53, 105

Offset: 0

Gus Wiseman, Nov 09 2018

Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different and none sums to 1.
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 weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 14 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                               {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{12}{13}{23}}  {{22}{11222}}
                                                            {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}

Cf. A000219, A007716, A059201, A302545, A316980, A316983, A319560, A319616.

Cf. A320796, A320797, A320798, A320804, A320811, A320812, A321401, A321404, A321405, A321406, A321407.

A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 3, 5, 9, 10, 22, 20, 37, 44, 65, 68, 127, 119, 182, 226, 307, 335, 511, 544, 782, 913, 1171, 1359, 1908, 2121, 2738, 3286, 4174, 4821, 6305, 7182, 9108, 10739, 13195, 15548, 19465, 22397, 27477, 32423, 39448, 45843, 55995, 64871, 78343, 91761, 109325

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of strict multiset partitions of constant multiset partitions of integer partitions of n.

			The a(1) = 1 through a(5) = 10 strict multiset partitions of constant multiset partitions of integer partitions:
  ((1))  ((2))     ((3))          ((4))             ((5))
         ((11))    ((21))         ((31))            ((41))
         ((1)(1))  ((111))        ((22))            ((32))
                   ((1)(1)(1))    ((211))           ((311))
                   ((1))((1)(1))  ((1111))          ((221))
                                  ((2)(2))          ((2111))
                                  ((11)(11))        ((11111))
                                  ((1)(1)(1)(1))    ((1)(1)(1)(1)(1))
                                  ((1))((1)(1)(1))  ((1))((1)(1)(1)(1))
                                                    ((1)(1))((1)(1)(1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A000041, A001970, A034729, A047968, A050343, A316980, A319066, A323764, A323766, A323774.

Join[{1}, Table[Sum[PartitionsQ[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1

Offset: 0

Gus Wiseman, Dec 28 2023

A set-system is a finite set of finite nonempty sets.

Conjecture: Column k = 2 is A101881.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   2   1
   0   1   4   3   1
   0   1   5   8   3   1
   0   1   8  18  13   3   1
   0   1   9  32  37  15   3   1
   0   1  13  55  96  59  16   3   1
   0   1  14  91 209 196  74  16   3   1
   0   1  19 138 449 573 313  82  16   3   1
   ...
Non-isomorphic representatives of the set-systems counted in row n = 5:
  .  {12345}  {1}{1234}  {1}{2}{123}  {1}{2}{3}{12}  {1}{2}{3}{4}{5}
              {1}{2345}  {1}{2}{134}  {1}{2}{3}{14}
              {12}{123}  {1}{2}{345}  {1}{2}{3}{45}
              {12}{134}  {1}{12}{13}
              {12}{345}  {1}{12}{23}
                         {1}{12}{34}
                         {1}{23}{24}
                         {1}{23}{45}

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

Row sums are A283877, connected case A300913.
For multiset partitions we have A317533.
Counting connected components instead of edges gives A321194.
For set multipartitions we have A334550.
For strict multiset partitions we have A368099.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A316980 counts non-isomorphic strict multiset partitions, connected A319557.
A319559 counts non-isomorphic T_0 set-systems, connected A319566.
Cf. A101881, A255903, A302545, A306005, A317532, A317794, A317795, A319560, A321405, A368094, A368095.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&And@@UnsameQ@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024

A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1

Offset: 0

Gus Wiseman, Dec 31 2023

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    4    1
    0    5   12    5    1
    0    7   28   22    5    1
    0   11   66   83   31    5    1
    0   15  134  252  147   34    5    1
    0   22  280  726  620  203   35    5    1
    0   30  536 1946 2283 1069  235   35    5    1
    0   42 1043 4982 7890 5019 1469  248   35    5    1
    ...
Row n = 4 counts the following representatives:
  .  {{1,1,1,1}}  {{1},{1,1,1}}  {{1},{2},{1,1}}  {{1},{2},{3},{4}}
     {{1,1,1,2}}  {{1},{1,1,2}}  {{1},{2},{1,2}}
     {{1,1,2,2}}  {{1},{1,2,2}}  {{1},{2},{1,3}}
     {{1,1,2,3}}  {{1},{1,2,3}}  {{1},{2},{3,3}}
     {{1,2,3,4}}  {{1},{2,2,2}}  {{1},{2},{3,4}}
                  {{1},{2,2,3}}
                  {{1},{2,3,4}}
                  {{1,1},{1,2}}
                  {{1,1},{2,2}}
                  {{1,1},{2,3}}
                  {{1,2},{1,3}}
                  {{1,2},{3,4}}

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

Row sums are A316980, connected case A319557.
For multiset partitions we have A317533, connected A322133.
Counting connected components instead of edges gives A321194.
For normal multiset partitions we have A330787, row sums A317776.
For set multipartitions we have A334550.
For set-systems we have A368096, row-sums A283877 (connected A300913).
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

A318357 Number of non-isomorphic strict multiset partitions of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 7, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 9, 1, 2, 3, 3, 2, 3, 1, 7, 2, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(48) = 7 strict multiset partitions of {1,1,1,1,2}:
  {{1,1,1,1,2}}
  {{1},{1,1,1,2}}
  {{2},{1,1,1,1}}
  {{1,1},{1,1,2}}
  {{1,2},{1,1,1}}
  {{1},{2},{1,1,1}}
  {{1},{1,1},{1,2}}

Cf. A001055, A007716, A045778, A056239, A112798, A316980, A317533, A317776, A317791, A318287.

A319625 Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 3

Offset: 0

Gus Wiseman, Sep 25 2018

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

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(10) = 3 antichains:
               {{1}}
        {{1,2},{1,3},{2,3}}
     {{1,2},{1,3},{2,4},{3,4}}
    {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,3},{2,4},{1,2,5},{3,4,5}}
  {{1,2},{1,3},{2,4},{3,5},{4,5}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646.

Euler transform is A319638.

A319628 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of (not necessarily distinct) multisets.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 10, 11, 37, 80, 233

Offset: 0

Gus Wiseman, Sep 25 2018

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

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,3}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,3,4}}
5: {{1,1,1,1,1}}
   {{1,2,3,4,5}}
   {{1,1},{1,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1,1,2,2,2}}
   {{1,1,2,2,3,3}}
   {{1,2,3,4,5,6}}
   {{1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2}}
   {{1,1,2},{2,2,2}}
   {{1,1,2},{2,3,3}}
   {{1,1},{1,2},{2,2}}
   {{1,2},{1,3},{2,3}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646.

Euler transform is A319641.

cp's OEIS Frontend

A340651 Number of non-isomorphic cross-balanced multiset partitions of weight n.

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A319751 Number of non-isomorphic set systems of weight n with empty intersection.

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A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

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A321402 Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.

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A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

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A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.

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A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

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A318357 Number of non-isomorphic strict multiset partitions of the multiset of prime indices of n.

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A319625 Number of non-isomorphic connected weight-n antichains of distinct sets whose dual is also an antichain of distinct sets.

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