A321155 -id:A321155 - cp's OEIS frontend

A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

0, 0, 0, 0, 1, 1, 5, 12, 36, 97, 291

Offset: 0

Gus Wiseman, Dec 23 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(5) = 1 through a(7) = 12 set-systems:
  {{1},{2},{3},{2,3}}  {{1},{2},{1,3},{2,3}}    {{1},{2},{1,2},{3,4,5}}
                       {{1},{2},{3},{1,2,3}}    {{1},{3},{2,3},{1,2,3}}
                       {{2},{3},{1,3},{2,3}}    {{1},{4},{1,4},{2,3,4}}
                       {{3},{4},{1,2},{3,4}}    {{2},{3},{2,3},{1,2,3}}
                       {{1},{2},{3},{4},{3,4}}  {{3},{1,2},{1,3},{2,3}}
                                                {{1},{2},{3},{1,3},{2,3}}
                                                {{1},{2},{3},{2,4},{3,4}}
                                                {{1},{2},{3},{4},{2,3,4}}
                                                {{1},{3},{4},{2,4},{3,4}}
                                                {{1},{4},{5},{2,3},{4,5}}
                                                {{2},{3},{4},{1,2},{3,4}}
                                                {{1},{2},{3},{4},{5},{4,5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
The case of labeled graphs is A367867, complement A133686.
The labeled version is A367903, ranks A367907.
The complement is counted by A368095, connected A368410.
Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368409.
Factorizations of this type are counted by A368413, complement A368414.
Allowing repeated edges gives A368421, complement A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A316983, A317533, A319616, A321155, A321405, A326031, A330223, A330227, A367905.

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@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,8}]

A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

Original entry on oeis.org

1, 3, 1, 6, 3, 1, 17, 12, 3, 1, 40, 35, 12, 3, 1, 125, 112, 45, 12, 3, 1, 354, 347, 148, 45, 12, 3, 1, 1159, 1122, 512, 163, 45, 12, 3, 1, 3774, 3651, 1724, 572, 163, 45, 12, 3, 1, 13113, 12320, 5937, 2020, 593, 163, 45, 12, 3, 1, 46426, 42407, 20492, 7117, 2110, 593, 163, 45, 12, 3, 1

Offset: 1

Gus Wiseman, Oct 29 2018

			Triangle begins:
      1
      3     1
      6     3     1
     17    12     3     1
     40    35    12     3     1
    125   112    45    12     3     1
    354   347   148    45    12     3     1
   1159  1122   512   163    45    12     3     1
   3774  3651  1724   572   163    45    12     3     1
  13113 12320  5937  2020   593   163    45    12     3     1
The fourth row counts the following non-isomorphic multiset partitions.
  {{1,1,1,1}}        {{1,1},{2,2}}      {{1},{2},{3,3}}    {{1},{2},{3},{4}}
  {{1,1,2,2}}        {{1},{2,2,2}}      {{1},{2},{3,4}}
  {{1,2,2,2}}        {{1},{2,3,3}}      {{1},{2},{3},{3}}
  {{1,2,3,3}}        {{1,2},{3,3}}
  {{1,2,3,4}}        {{1},{2,3,4}}
  {{1},{1,1,1}}      {{1,2},{3,4}}
  {{1,1},{1,1}}      {{1},{1},{2,2}}
  {{1},{1,2,2}}      {{1},{1},{2,3}}
  {{1,2},{1,2}}      {{1},{2},{2,2}}
  {{1,2},{2,2}}      {{1},{3},{2,3}}
  {{1,3},{2,3}}      {{1},{1},{2},{2}}
  {{2},{1,2,2}}      {{1},{2},{2},{2}}
  {{3},{1,2,3}}
  {{1},{1},{1,1}}
  {{1},{2},{1,2}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

First column is A007718. Row sums are A007716.

Cf. A006126, A048143, A143543, A306005, A316983, A317672, A317674, A319616, A321155.

O.g.f.: Product 1/(1 - t*x^n)^A007718(n).

Terms a(56) and beyond from Andrew Howroyd, Jan 11 2024

A321229 Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.

Original entry on oeis.org

1, 1, 3, 6, 16, 37, 105, 279, 817, 2387, 7269

Offset: 0

Gus Wiseman, Oct 31 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

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(5) = 37 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}        {{1,1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}        {{1,2,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}        {{1,2,2,3,3}}
                    {{2},{1,2}}    {{1,2,3,4}}        {{1,2,3,3,3}}
                    {{1},{1},{1}}  {{1},{1,1,1}}      {{1,2,3,4,4}}
                                   {{1,1},{1,1}}      {{1,2,3,4,5}}
                                   {{1},{1,2,2}}      {{1},{1,1,1,1}}
                                   {{1,2},{2,2}}      {{1,1},{1,1,1}}
                                   {{1,3},{2,3}}      {{1,1},{1,2,2}}
                                   {{2},{1,2,2}}      {{1},{1,2,2,2}}
                                   {{3},{1,2,3}}      {{1,2},{2,2,2}}
                                   {{1},{1},{1,1}}    {{1,2},{2,3,3}}
                                   {{1},{2},{1,2}}    {{1,3},{2,3,3}}
                                   {{2},{2},{1,2}}    {{1,4},{2,3,4}}
                                   {{1},{1},{1},{1}}  {{2},{1,1,2,2}}
                                                      {{2},{1,2,2,2}}
                                                      {{2},{1,2,3,3}}
                                                      {{2,2},{1,2,2}}
                                                      {{3},{1,2,3,3}}
                                                      {{3,3},{1,2,3}}
                                                      {{4},{1,2,3,4}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1,2,2}}
                                                      {{1},{1,2},{2,2}}
                                                      {{1},{2},{1,2,2}}
                                                      {{2},{1,2},{2,2}}
                                                      {{2},{1,3},{2,3}}
                                                      {{2},{2},{1,2,2}}
                                                      {{2},{3},{1,2,3}}
                                                      {{3},{1,3},{2,3}}
                                                      {{3},{3},{1,2,3}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321227, A321228, A321231.

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

Original entry on oeis.org

1, 3, 0, 6, 0, 0, 16, 1, 0, 0, 37, 3, 0, 0, 0, 105, 18, 2, 0, 0, 0, 279, 68, 7, 0, 0, 0, 0, 817, 293, 46, 3, 0, 0, 0, 0, 2387, 1141, 228, 17, 1, 0, 0, 0, 0, 7269, 4511, 1189, 135, 9, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			Triangle begins:
     1
     3    0
     6    0    0
    16    1    0    0
    37    3    0    0    0
   105   18    2    0    0    0
   279   68    7    0    0    0    0
   817  293   46    3    0    0    0    0
  2387 1141  228   17    1    0    0    0    0
  7269 4511 1189  135    9    0    0    0    0    0

First column is A321229. Row sums are A007718.

Cf. A007716, A317672, A321155, A321194, A321253, A321255, A321256.

A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 9, 24, 51, 134, 328, 868

Offset: 1

Gus Wiseman, Nov 29 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

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(6) = 24 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}
          {{123}}  {{1222}}    {{12222}}    {{112222}}
                   {{1233}}    {{12233}}    {{112233}}
                   {{1234}}    {{12333}}    {{122222}}
                   {{13}{23}}  {{12344}}    {{122333}}
                               {{12345}}    {{123333}}
                               {{12}{233}}  {{123344}}
                               {{13}{233}}  {{123444}}
                               {{14}{234}}  {{123455}}
                                            {{123456}}
                                            {{112}{233}}
                                            {{122}{233}}
                                            {{12}{2333}}
                                            {{123}{344}}
                                            {{124}{344}}
                                            {{125}{345}}
                                            {{13}{2233}}
                                            {{13}{2333}}
                                            {{13}{2344}}
                                            {{133}{233}}
                                            {{14}{2344}}
                                            {{15}{2345}}
                                            {{13}{24}{34}}
                                            {{14}{24}{34}}

Cf. A007716, A007718, A286520, A304867, A319719, A319721, A321155, A321760, A322111, A322113.

A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where the multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices. - Gus Wiseman, Nov 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 2 through a(5) = 35 connected multigraphs with loops with multiset density -1.
Index entries for sequences related to trees

Essentially the same as A036251.

Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.

max = 30; B[] = 1; Do[B[x] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.

A125702 Number of connected categories with n objects and 2n-1 morphisms.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793

Offset: 1

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

nonn

Also number of connected antitransitive relations on n objects (antitransitive meaning a R b and b R c implies not a R c); equivalently, number of free oriented bipartite trees, with all arrows going from one part to the other part.

Also the number of non-isomorphic multi-hypertrees of weight n - 1 with singletons allowed. A multi-hypertree with singletons allowed is a connected set multipartition (multiset of sets) with density -1, where the density of a set multipartition is the weight (sum of sizes of the parts) minus the number of parts minus the number of vertices. - Gus Wiseman, Oct 30 2018

			From _Gus Wiseman_, Oct 30 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees of weight n - 1 with singletons allowed:
  {}  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
             {{1}{1}}  {{2}{12}}    {{13}{23}}      {{14}{234}}
                       {{1}{1}{1}}  {{3}{123}}      {{4}{1234}}
                                    {{1}{2}{12}}    {{2}{13}{23}}
                                    {{2}{2}{12}}    {{2}{3}{123}}
                                    {{1}{1}{1}{1}}  {{3}{13}{23}}
                                                    {{3}{3}{123}}
                                                    {{1}{2}{2}{12}}
                                                    {{2}{2}{2}{12}}
                                                    {{1}{1}{1}{1}{1}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..500

Same as A122086 except for n = 1; see there for formulas. Cf. A125699.

Cf. A000081, A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A318697, A320921, A321155.

\\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2 - x)} \\ Andrew Howroyd, Nov 02 2019

a(n) = A122086(n) for n > 1.

G.f.: 2*f(x) - f(x)^2 - x where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019

A321228 Number of non-isomorphic hypertrees of weight n with singletons.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 13, 23, 49, 100, 220

Offset: 0

Gus Wiseman, Oct 31 2018

A hypertree with singletons is a connected set system (finite set of finite nonempty sets) with density -1, where the density of a set system is the sum of sizes of the parts (weight) minus the number of parts minus the number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 23 hypertrees:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{2},{1,2}}  {{1,3},{2,3}}    {{1,4},{2,3,4}}
                               {{3},{1,2,3}}    {{4},{1,2,3,4}}
                               {{1},{2},{1,2}}  {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}
.
  {{1,2,3,4,5,6}}        {{1,2,3,4,5,6,7}}
  {{1,2,5},{3,4,5}}      {{1,2,6},{3,4,5,6}}
  {{1,5},{2,3,4,5}}      {{1,6},{2,3,4,5,6}}
  {{5},{1,2,3,4,5}}      {{6},{1,2,3,4,5,6}}
  {{1},{1,4},{2,3,4}}    {{1},{1,5},{2,3,4,5}}
  {{1,3},{2,4},{3,4}}    {{1,2},{2,5},{3,4,5}}
  {{1,4},{2,4},{3,4}}    {{1,4},{2,5},{3,4,5}}
  {{3},{1,4},{2,3,4}}    {{1,5},{2,5},{3,4,5}}
  {{3},{4},{1,2,3,4}}    {{4},{1,2,5},{3,4,5}}
  {{4},{1,4},{2,3,4}}    {{4},{1,5},{2,3,4,5}}
  {{1},{2},{1,3},{2,3}}  {{4},{5},{1,2,3,4,5}}
  {{1},{2},{3},{1,2,3}}  {{5},{1,2,5},{3,4,5}}
  {{2},{3},{1,3},{2,3}}  {{5},{1,5},{2,3,4,5}}
                         {{1},{3},{1,4},{2,3,4}}
                         {{1},{4},{1,4},{2,3,4}}
                         {{2},{1,3},{2,4},{3,4}}
                         {{2},{3},{1,4},{2,3,4}}
                         {{2},{3},{4},{1,2,3,4}}
                         {{3},{1,4},{2,4},{3,4}}
                         {{3},{4},{1,4},{2,3,4}}
                         {{4},{1,3},{2,4},{3,4}}
                         {{4},{1,4},{2,4},{3,4}}
                         {{1},{2},{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321227, A321229.

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 12, 28, 78, 202, 578, 1650, 4904

Offset: 0

Gus Wiseman, Nov 01 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

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(5) = 28 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}      {{1,1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}      {{1,1,2,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}      {{1,2,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}      {{1,2,2,3,3}}
                  {{2},{1,2}}  {{1,2,3,4}}      {{1,2,3,3,3}}
                               {{1},{1,1,1}}    {{1,2,3,4,4}}
                               {{1},{1,2,2}}    {{1,2,3,4,5}}
                               {{1,2},{2,2}}    {{1},{1,1,1,1}}
                               {{1,3},{2,3}}    {{1,1},{1,1,1}}
                               {{2},{1,2,2}}    {{1,1},{1,2,2}}
                               {{3},{1,2,3}}    {{1},{1,2,2,2}}
                               {{1},{2},{1,2}}  {{1,2},{2,2,2}}
                                                {{1,2},{2,3,3}}
                                                {{1,3},{2,3,3}}
                                                {{1,4},{2,3,4}}
                                                {{2},{1,1,2,2}}
                                                {{2},{1,2,2,2}}
                                                {{2},{1,2,3,3}}
                                                {{2,2},{1,2,2}}
                                                {{3},{1,2,3,3}}
                                                {{3,3},{1,2,3}}
                                                {{4},{1,2,3,4}}
                                                {{1},{1,2},{2,2}}
                                                {{1},{2},{1,2,2}}
                                                {{2},{1,2},{2,2}}
                                                {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321194, A321228, A321229, A321254.

A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 13, 13, 37, 37

Offset: 0

Gus Wiseman, Nov 26 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
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}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
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(8) = 13 multiset partitions:
  {{1}}                    {{1,1}}
.
  {{1,1,1}}                {{1,1,1,1}}
  {{2},{1,2}}              {{2},{1,2,2}}
.
  {{1,1,1,1,1}}            {{1,1,1,1,1,1}}
  {{1,1},{1,2,2}}          {{2},{1,2,2,2,2}}
  {{2},{1,2,2,2}}          {{2,2},{1,1,2,2}}
  {{2},{1,3},{2,3}}        {{2},{1,3},{2,3,3}}
  {{3},{3},{1,2,3}}        {{3},{3},{1,2,3,3}}
.
  {{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},{1,1,2,2,2}}
  {{2},{1,2,2,2,2,2}}      {{2},{1,2,2,2,2,2,2}}
  {{2,2},{1,1,2,2,2}}      {{2,2},{1,1,2,2,2,2}}
  {{1,1},{1,2},{2,3,3}}    {{1,1},{1,2,2},{2,3,3}}
  {{2},{1,3},{2,3,3,3}}    {{2},{1,3},{2,3,3,3,3}}
  {{2},{2,2},{1,2,3,3}}    {{2},{1,3,3},{2,2,3,3}}
  {{3},{1,2,2},{2,3,3}}    {{3},{3},{1,2,3,3,3,3}}
  {{3},{3},{1,2,3,3,3}}    {{3},{3,3},{1,2,2,3,3}}
  {{1},{1},{1,4},{2,3,4}}  {{2},{1,3},{2,4},{3,4,4}}
  {{2},{1,3},{2,4},{3,4}}  {{3},{3},{1,2,4},{3,4,4}}
  {{3},{4},{1,4},{2,3,4}}  {{3},{4},{1,4},{2,3,4,4}}
  {{4},{4},{4},{1,2,3,4}}  {{4},{4},{4},{1,2,3,4,4}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A318697, A319616, A321155, A321255.

cp's OEIS Frontend

A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

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A321194 Regular triangle where T(n,k) is the number of non-isomorphic multiset partitions of weight n with k connected components.

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A321229 Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.

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A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

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A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

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A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A125702 Number of connected categories with n objects and 2n-1 morphisms.

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A321228 Number of non-isomorphic hypertrees of weight n with singletons.

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A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

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A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

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