A283877 -id:A283877 - cp's OEIS frontend

A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

1, 1, 3, 29, 1885, 18658259

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 T_0 condition means the dual is strict (no repeated elements).

			Non-isomorphic representatives of the a(3) = 29 covers:
   {{1,3},{2,3}}
   {{1},{2},{3}}
   {{1},{3},{2,3}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{3},{1,3},{2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2,3}}
   {{1},{2},{1,3},{2,3}}
   {{2},{3},{1,3},{2,3}}
   {{1},{3},{2,3},{1,2,3}}
   {{2},{3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3}}
   {{2},{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,3},{2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{3},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A316980, A316983, A318099, A319558, A319559, A319616-A319646, A300913.

a(5) from Max Alekseyev, Jul 13 2022

A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 7, 8, 15, 18, 30, 37, 58, 71, 105, 131, 186, 230, 318, 393, 530, 653, 863, 1060, 1380, 1686, 2164, 2637, 3345, 4057, 5096, 6158, 7665, 9228, 11395, 13671, 16765, 20040, 24418, 29098, 35251, 41869, 50460, 59755, 71669, 84626, 101050

Offset: 0

Henry Bottomley, Aug 01 2005

nonn

Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018

			a(5) = 3 since 5 can be partitioned as 1+1+1+1+1, 2+1+1+1, or 2+2+1; not counted are 5, 4+1, or 3+2.
a(6) = 7 since 6 can be partitioned as 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+1+3, 1+2+3, 3+3; not counted are 1+1+4, 2+4, 1+5, 6.
From _Gus Wiseman_, Oct 30 2018: (Start)
The a(2) = 1 through a(8) = 15 partitions with no part larger than n/2:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (2111)   (222)     (331)      (332)
               (1111)  (11111)  (321)     (2221)     (422)
                                (2211)    (3211)     (431)
                                (3111)    (22111)    (2222)
                                (21111)   (31111)    (3221)
                                (111111)  (211111)   (3311)
                                          (1111111)  (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(2) = 1 through a(8) = 15 partitions into n/2 or fewer parts:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (22)  (32)  (33)   (43)   (44)
            (31)  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (222)  (322)  (71)
                        (321)  (331)  (332)
                        (411)  (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (2222)
                                      (3221)
                                      (3311)
                                      (4211)
                                      (5111)
The a(6) = 7 integer partitions of 6 with no part larger than n/2 together with a realizing set multipartition of each (the parts of the partition count the appearances of each vertex in the set multipartition):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2},{1,2,3,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}
(End)

Cf. A000070, A000569, A025065, A049311, A096373, A116540, A147878, A209816, A283877, A306005, A320921.

A000070 := proc(n) add( combinat[numbpart](i),i=0..n) ; end proc:
A110618 := proc(n) combinat[numbpart](n) - A000070(floor((n-1)/2)) ; end proc: # R. J. Mathar, Jan 24 2011

f[n_, 1] := 1; f[1, k_] := 1; f[n_, k_] := f[n, k] = If[k > n, f[n, k - 1], f[n, k - 1] + f[n - k, k]]; g[n_] := f[n, Floor[n/2]]; g[0] = 1; g[1] = 0; Array[g, 47, 0] (* Robert G. Wilson v, Jan 23 2011 *)
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]]]];
multhyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multhyp[#]!={}&]],{n,8}] (* Gus Wiseman, Oct 30 2018 *)

a(n) = numbpart(n) - sum(i=0, if (n%2, n\2, n/2-1), numbpart(i)); \\ Michel Marcus, Oct 31 2018

a(n) = A000041(n) - Sum_{i=0..floor((n-1)/2)} A000041(i) = A000041(n) - A000070(floor((n-1)/2)) = A110619(n, 2).

a(2*n) = A209816(n). - Gus Wiseman, Oct 30 2018

A321405 Number of non-isomorphic self-dual set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 9, 16, 28, 47

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of (0,1) 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.
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(1) = 1 through a(8) = 16 set systems:
  {{1}}  {{1}{2}}  {{2}{12}}    {{1}{3}{23}}    {{2}{13}{23}}
                   {{1}{2}{3}}  {{1}{2}{3}{4}}  {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}
.
  {{12}{13}{23}}        {{13}{23}{123}}          {{1}{13}{14}{234}}
  {{3}{23}{123}}        {{1}{23}{24}{34}}        {{12}{13}{24}{34}}
  {{1}{3}{24}{34}}      {{1}{4}{34}{234}}        {{1}{24}{34}{234}}
  {{2}{4}{12}{34}}      {{2}{13}{24}{34}}        {{2}{14}{34}{234}}
  {{1}{2}{3}{5}{45}}    {{3}{4}{14}{234}}        {{3}{4}{134}{234}}
  {{1}{2}{3}{4}{5}{6}}  {{1}{2}{4}{35}{45}}      {{4}{13}{14}{234}}
                        {{1}{3}{5}{23}{45}}      {{1}{2}{34}{35}{45}}
                        {{1}{2}{3}{4}{6}{56}}    {{1}{2}{5}{45}{345}}
                        {{1}{2}{3}{4}{5}{6}{7}}  {{1}{3}{24}{35}{45}}
                                                 {{1}{4}{5}{25}{345}}
                                                 {{2}{4}{12}{35}{45}}
                                                 {{4}{5}{13}{23}{45}}
                                                 {{1}{2}{3}{5}{46}{56}}
                                                 {{1}{2}{4}{6}{34}{56}}
                                                 {{1}{2}{3}{4}{5}{7}{67}}
                                                 {{1}{2}{3}{4}{5}{6}{7}{8}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A316980, A316983, A319616.

Cf. A320796, A320797, A321401, A321403, A321406.

A292432 Number of normal multisets that cannot be expressed as the multiset-union of a set of sets.

Original entry on oeis.org

0, 1, 1, 3, 5, 9, 16, 27, 46, 76, 130, 203, 350, 554, 890, 1474, 2285, 3732, 5852, 9297, 14628, 22936, 35903, 55893, 86967, 134585, 207934, 321122, 492634, 757490

Offset: 1

Gus Wiseman, Oct 02 2017

A multiset is normal if it spans an initial interval of positive integers. Most normal multisets can be expressed as the multiset-union of a set of sets. For example, {1,1,2,2} is the multiset-union of {{1},{2},{1,2}}.

			The a(6) = 9 multisets are: {1,1,1,1,1,1}, {1,1,1,1,1,2}, {1,1,1,1,2,2}, {1,1,1,1,2,3}, {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}.

Cf. A049311, A089259, A116540, A283877, A292444.

a(11)-a(30) from Bert Dobbelaere, Mar 30 2025

A319765 Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 74, 156, 358, 792, 1821

Offset: 0

Gus Wiseman, Sep 27 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}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
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(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1},{1},{1,1}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319766, A319767, A319768, A319769, A319773, A319774.

A292444 Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 30, 39, 50, 65, 83, 105, 131, 167, 207, 257, 317, 391, 478, 585, 708, 864, 1037, 1252, 1498

Offset: 1

Gus Wiseman, Oct 02 2017

Non-isomorphic finite multisets correspond to integer partitions. For example, the partition (3221) corresponds to the multiset {1,1,1,2,2,3,3,4}.

			Representatives of the a(7) = 6 multisets are: {1,1,1,1,1,1,1}, {1,1,1,1,1,1,2}, {1,1,1,1,1,2,2}, {1,1,1,1,1,2,3}, {1,1,1,1,2,2,2}, {1,1,1,1,2,2,3}.

Cf. A049311, A089259, A116540, A283877, A292432.

a(12)-a(30) from Bert Dobbelaere, Mar 30 2025

A293510 Number of connected minimal covers of n vertices.

Original entry on oeis.org

1, 1, 1, 4, 23, 241, 3732, 83987, 2666729, 117807298, 7217946453, 612089089261, 71991021616582, 11761139981560581, 2675674695560997301, 849270038176762472316, 376910699272413914514283, 234289022942841270608166061, 204344856617470777364053906796

Offset: 0

Gus Wiseman, Oct 11 2017

nonn

A cover of a finite set S is a finite set of finite nonempty sets with union S. A cover is minimal if removing any edge results in a cover of strictly fewer vertices. A cover is connected if it is connected as a hypergraph or clutter. Note that minimality is with respect to covering rather than to connectedness (cf. A030019).

			The a(3) = 4 covers are: ((12)(13)), ((12)(23)), ((13)(23)), ((123)).

Cf. A030019, A046165, A048143, A275307, A283877.

nn=30;ser=Sum[(1+Sum[Binomial[n,i]*StirlingS2[i,k]*(2^k-k-1)^(n-i),{k,2,n},{i,k,n}])*x^n/n!,{n,0,nn}];
Table[n!*SeriesCoefficient[1+Log[ser],{x,0,n}],{n,0,nn}]

A319190 Number of regular hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 3, 19, 879, 5280907, 1069418570520767

Offset: 0

Gus Wiseman, Dec 17 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

			The a(3) = 19 regular hypergraphs:
                 {{1,2,3}}
                {{1},{2,3}}
                {{2},{1,3}}
                {{3},{1,2}}
               {{1},{2},{3}}
            {{1},{2,3},{1,2,3}}
            {{2},{1,3},{1,2,3}}
            {{3},{1,2},{1,2,3}}
            {{1,2},{1,3},{2,3}}
           {{1},{2},{3},{1,2,3}}
           {{1},{2},{1,3},{2,3}}
           {{1},{3},{1,2},{2,3}}
           {{2},{3},{1,2},{1,3}}
        {{1,2},{1,3},{2,3},{1,2,3}}
       {{1},{2},{1,3},{2,3},{1,2,3}}
       {{1},{3},{1,2},{2,3},{1,2,3}}
       {{2},{3},{1,2},{1,3},{1,2,3}}
      {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Column sums of A188445.

Cf. A002829, A005176, A049311, A058891, A110100, A110101, A116539, A283877, A295193, A306017, A319189.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{1,n}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,2^n}],{n,5}]

a(6) from Andrew Howroyd, Mar 12 2020

A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 4, 8, 21, 62, 175, 553, 1775, 6007

Offset: 0

Gus Wiseman, Sep 23 2018

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent 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(4) = 8 multiset partitions:
1:      {{1}}
2:     {{1,1}}
3:    {{1,1,1}}
      {{1,2,2}}
     {{1},{1,1}}
     {{2},{1,2}}
4:   {{1,1,1,1}}
     {{1,2,2,2}}
    {{1},{1,1,1}}
    {{1},{1,2,2}}
    {{2},{1,2,2}}
    {{1,2},{2,2}}
    {{1,3},{2,3}}
   {{1},{2},{1,2}}

Cf. A007716, A007718, A049311, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319560, A319564, A319566, A319567.

A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 1, 1490, 67027582, 144115188036455750, 1329227995784915872903806998967001298, 226156424291633194186662080095093570025917938800079226639565284090686126876

Offset: 0

Gus Wiseman, Dec 05 2023

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.

Includes all set-systems with more edges than covered vertices, but this condition is not sufficient.

			The a(3) = 1 set-system is: {{1,2},{1,3},{2,3},{1,2,3}}.

Wikipedia, Axiom of choice.

Set-systems without singletons are counted by A016031, covering A323816.
The complement is A367770, with singletons allowed A367902 (ranks A367906).
The version for simple graphs is A367867, covering A367868.
The version allowing singletons and empty edges is A367901.
The version allowing singletons is A367903, ranks A367907.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
Cf. A007716, A092918, A102896, A283877, A306445, A326031, A355739, A355740, A355741, A367904, A367905.

Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n,0,3}]

a(n) = 2^(2^n-n-1) - A367770(n) = A016031(n+1) - A367770(n). - Christian Sievers, Jul 28 2024

a(6)-a(8) from Christian Sievers, Jul 28 2024

cp's OEIS Frontend

A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

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A110618 Number of partitions of n with no part larger than n/2. Also partitions of n into n/2 or fewer parts.

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A321405 Number of non-isomorphic self-dual set systems of weight n.

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A292432 Number of normal multisets that cannot be expressed as the multiset-union of a set of sets.

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A319765 Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.

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A292444 Number of non-isomorphic finite multisets that cannot be expressed as the multiset-union of a set of sets.

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A293510 Number of connected minimal covers of n vertices.

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Mathematica

A319190 Number of regular hypergraphs spanning n vertices.

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Mathematica

Extensions

A319565 Number of non-isomorphic connected strict T_0 multiset partitions of weight n.

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A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

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