A299353 -id:A299353 - cp's OEIS frontend

A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.

1, 1, 2, 6, 54, 1754, 1102746, 68715913086, 1180735735356265746734, 170141183460507906731293351306656207090, 7237005577335553223087828975127304177495735363998991435497132232365910414322

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024

			The a(3) = 6 set-systems in which all sets have the same size:
  {{1,2,3}}
  {{1}, {2}, {3}}
  {{1,2}, {1,3}}
  {{1,2}, {2,3}}
  {{1,3}, {2,3}}
  {{1,2}, {1,3}, {2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..14

Row sums of A299471.
The unlabeled version is A301481.
The connected version is A299353.
Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020.

Table[Sum[(-1)^(n-k)*Binomial[n,k]*(1+Sum[2^Binomial[k,d]-1,{d,k}]),{k,0,n}],{n,12}]

a(n) = if(n==0, 1, sum(k=0, n, sum(d=0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)))) \\ Andrew Howroyd, Jan 16 2024

a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).

Inverse binomial transform of A306020. - Andrew Howroyd, Jan 16 2024

A295193 Number of regular simple graphs on n labeled nodes.

Original entry on oeis.org

1, 2, 2, 8, 14, 172, 932, 45936, 1084414, 155862512, 10382960972, 6939278572096, 2203360500122300, 4186526756621772344, 3747344008241368443820, 35041787059691023579970848, 156277111373303386104606663422, 4142122641757598618318165240180096

Offset: 1

Álvar Ibeas, Nov 16 2017

nonn

			From _Gus Wiseman_, Dec 19 2018: (Start)
A graph is regular if all vertices have the same degree. For example, the a(4) = 8 simple regular graphs are:
  1 2
  3 4
.
  4---1  3---1  2---1
  3---2  4---2  4---3
.
  3---4  4---3  4---2
  |   |  |   |  |   |
  1---2  1---2  1---3
.
  4---3
  | X |
  2---1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..24
E. A. Bender and E. R. Canfield, The asymptotic number of labeled graphs with given degree sequences, Journal of Combinatorial Theory, Series A, 24 (1978), 296-307.
Andrew Howroyd, PARI Program
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
B. D. McKay, Applications of a technique for labelled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Wikipedia, Regular graph

Row sums of A059441.

Cf. A005176 (unlabeled equivalent), A058891, A116539, A299353, A306017, A306021, A319189, A319190, A319612, A319729.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,9}] (* Gus Wiseman, Dec 19 2018 *)

\\ See link for program file.
for(n=1, 10, print1(A295193(n), ", ")) \\ Andrew Howroyd, Aug 28 2019

a(16)-a(18) from Andrew Howroyd, Aug 28 2019

A319189 Number of uniform regular hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 2, 3, 10, 29, 3780, 5012107

Offset: 0

Gus Wiseman, Dec 17 2018

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

Also the number of 0-1 matrices with n columns, all distinct rows, no zero columns, equal row-sums, and equal column-sums, up to a permutation of the rows.

			The a(4) = 10 edge-sets:
               {{1,2,3,4}}
              {{1,2},{3,4}}
              {{1,3},{2,4}}
              {{1,4},{2,3}}
            {{1},{2},{3},{4}}
        {{1,2},{1,3},{2,4},{3,4}}
        {{1,2},{1,4},{2,3},{3,4}}
        {{1,3},{1,4},{2,3},{2,4}}
    {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
Inequivalent representatives of the a(4) = 10 matrices:
  [1 1 1 1]
.
  [1 1 0 0] [1 0 1 0] [1 0 0 1]
  [0 0 1 1] [0 1 0 1] [0 1 1 0]
.
  [1 0 0 0] [1 1 0 0] [1 1 0 0] [1 0 1 0] [1 1 1 0]
  [0 1 0 0] [1 0 1 0] [1 0 0 1] [1 0 0 1] [1 1 0 1]
  [0 0 1 0] [0 1 0 1] [0 1 1 0] [0 1 1 0] [1 0 1 1]
  [0 0 0 1] [0 0 1 1] [0 0 1 1] [0 1 0 1] [0 1 1 1]
.
  [1 1 0 0]
  [1 0 1 0]
  [1 0 0 1]
  [0 1 1 0]
  [0 1 0 1]
  [0 0 1 1]

Uniform hypergraphs are counted by A306021. Unlabeled uniform regular multiset partitions are counted by A319056. Regular graphs are A295193. Uniform clutters are A299353.

Cf. A002829, A005176, A007016, A049311, A058891, A101370, A110100, A110101, A321717, A321720.

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

a(7) from Jinyuan Wang, Jun 20 2020

A322635 Number of regular graphs with loops on n labeled vertices.

Original entry on oeis.org

2, 4, 4, 24, 78, 1908, 23368, 1961200, 75942758, 25703384940, 4184912454930, 4462909435830552, 2245354417775573206, 10567193418810168583576, 24001585002447984453495392, 348615956932626441906675011568, 2412972383955442904868321667433106, 162906453913051798826796439651249753404

Offset: 1

Gus Wiseman, Dec 21 2018

nonn

A graph is regular if all vertices have the same degree. A loop adds 2 to the degree of its vertex.

Wikipedia, Regular graph
Gus Wiseman, The a(4) = 24 regular graphs with loops.
Gus Wiseman, The a(5) = 78 regular graphs with loops.

Row sums of A333158.

Cf. A000666, A054921, A059441, A295193, A299353, A306017, A306021, A319189, A319190, A319612, A322659, A322661.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2n}],{n,6}]

for(n=1, 10, print1(A322635(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(11)-a(18) from Andrew Howroyd, Aug 28 2019

A322785 Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 4, 12, 4, 48, 4, 183, 297, 1186, 4, 33950, 4, 139527, 1529608, 4726356, 4, 229255536, 4, 3705777010, 36279746314, 13764663019, 4, 14096735197959, 5194673049514, 7907992957755, 2977586461058927, 13426396910491001, 4, 1350012288268171854, 4, 59487352224070807287

Offset: 0

Gus Wiseman, Dec 26 2018

nonn

A multiset is uniform if all multiplicities are equal. A multiset partition is uniform if all parts have the same size.

			The a(1) = 1 though a(6) = 48 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}          {111111}
       {12}    {123}      {1122}        {12345}          {111222}
       {1}{1}  {1}{1}{1}  {1234}        {1}{1}{1}{1}{1}  {112233}
       {1}{2}  {1}{2}{3}  {11}{11}      {1}{2}{3}{4}{5}  {123456}
                          {11}{22}                       {111}{111}
                          {12}{12}                       {111}{222}
                          {12}{34}                       {112}{122}
                          {13}{24}                       {112}{233}
                          {14}{23}                       {113}{223}
                          {1}{1}{1}{1}                   {122}{133}
                          {1}{1}{2}{2}                   {123}{123}
                          {1}{2}{3}{4}                   {123}{456}
                                                         {124}{356}
                                                         {125}{346}
                                                         {126}{345}
                                                         {134}{256}
                                                         {135}{246}
                                                         {136}{245}
                                                         {145}{236}
                                                         {146}{235}
                                                         {156}{234}
                                                         {11}{11}{11}
                                                         {11}{12}{22}
                                                         {11}{22}{33}
                                                         {11}{23}{23}
                                                         {12}{12}{12}
                                                         {12}{12}{33}
                                                         {12}{13}{23}
                                                         {12}{34}{56}
                                                         {12}{35}{46}
                                                         {12}{36}{45}
                                                         {13}{13}{22}
                                                         {13}{24}{56}
                                                         {13}{25}{46}
                                                         {13}{26}{45}
                                                         {14}{23}{56}
                                                         {14}{25}{36}
                                                         {14}{26}{35}
                                                         {15}{23}{46}
                                                         {15}{24}{36}
                                                         {15}{26}{34}
                                                         {16}{23}{45}
                                                         {16}{24}{35}
                                                         {16}{25}{34}
                                                         {1}{1}{1}{1}{1}{1}
                                                         {1}{1}{1}{2}{2}{2}
                                                         {1}{1}{2}{2}{3}{3}
                                                         {1}{2}{3}{4}{5}{6}

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row sums of A322788.

Cf. A000040, A038041, A072774, A100778, A299353, A306017, A306018, A306021, A317583, A317584, A319056, A319189, A321721, A322705, A322784, A322788.

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[m],SameQ@@Length/@#&]],{m,Table[Join@@Table[Range[n/d],{d}],{d,Divisors[n]}]}],{n,8}]

a(n) = 4 <=> n in { A000040 }. - Alois P. Heinz, Feb 03 2022

More terms from Alois P. Heinz, Jan 30 2019

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2022

A301920 Number of unlabeled uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 3, 10, 55, 2369, 14026242, 29284932065996223, 468863491068204425232922367146585, 1994324729204021501147398087008429476673379600542622915802043455294332

Offset: 0

Gus Wiseman, Jun 19 2018

nonn

A hypergraph is uniform if all edges have the same size.

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

Andrew Howroyd, Table of n, a(n) for n = 0..13

Row sums of A301924.

Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A301481, A306017-A306021.

Terms a(6) and beyond from Andrew Howroyd, Aug 26 2019

A299354 Regular triangle where T(n,k) is the number of labeled connected k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 38, 11, 1, 0, 728, 958, 26, 1, 0, 26704, 1042632, 32596, 57, 1, 0, 1866256, 34352418950, 34359509614, 2096731, 120, 1, 0, 251548592, 72057319189266922, 1180591620442534312262, 72057594021152435, 268434467, 247, 1, 0, 66296291072

Offset: 1

Gus Wiseman, Jun 18 2018

			Triangle begins:
1
0, 1
0, 4, 1
0, 38, 11, 1
0, 728, 958, 26, 1
0, 26704, 1042632, 32596, 57, 1

Wikipedia, Hypergraph

Second column is A001187. Row sums are A299353.

Cf. A000005, A001315, A006126, A006129, A038041, A048143, A298422, A299353, A299471, A306020, A306021.

nn=10;Table[SeriesCoefficient[Log[Sum[x^n/n!*Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,0,nn}]],{x,0,n}]*n!,{n,nn},{k,n}]

Column k is the logarithmic transform of the inverse binomial transform of c(d) = 2^binomial(d,k).

A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 41, 43, 47, 49, 51, 53, 55, 59, 64, 67, 73, 79, 81, 83, 85, 93, 97, 101, 103, 109, 113, 121, 123, 125, 127, 128, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 169, 177, 179

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform and regular, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems, together with their MM-numbers, begins:
   1: {}                   33: {{1},{3}}            109: {{10}}
   2: {{}}                 41: {{6}}                113: {{1,2,3}}
   3: {{1}}                43: {{1,4}}              121: {{3},{3}}
   4: {{},{}}              47: {{2,3}}              123: {{1},{6}}
   5: {{2}}                49: {{1,1},{1,1}}        125: {{2},{2},{2}}
   7: {{1,1}}              51: {{1},{4}}            127: {{11}}
   8: {{},{},{}}           53: {{1,1,1,1}}          128: {{},{},{},{},{},{}}
   9: {{1},{1}}            55: {{2},{3}}            131: {{1,1,1,1,1}}
  11: {{3}}                59: {{7}}                137: {{2,5}}
  13: {{1,2}}              64: {{},{},{},{},{},{}}  139: {{1,7}}
  15: {{1},{2}}            67: {{8}}                149: {{3,4}}
  16: {{},{},{},{}}        73: {{2,4}}              151: {{1,1,2,2}}
  17: {{4}}                79: {{1,5}}              155: {{2},{5}}
  19: {{1,1,1}}            81: {{1},{1},{1},{1}}    157: {{12}}
  23: {{2,2}}              83: {{9}}                161: {{1,1},{2,2}}
  25: {{2},{2}}            85: {{2},{4}}            163: {{1,8}}
  27: {{1},{1},{1}}        93: {{1},{5}}            165: {{1},{2},{3}}
  29: {{1,3}}              97: {{3,3}}              167: {{2,6}}
  31: {{5}}               101: {{1,6}}              169: {{1,2},{1,2}}
  32: {{},{},{},{},{}}    103: {{2,2,2}}            177: {{1},{7}}

Cf. A005176, A007016, A112798, A271103, A283877, A299353, A302242, A306017, A319056, A319189, A320324, A321699, A321717, A322554, A322703, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322659 Number of connected regular simple graphs on n labeled vertices.

Original entry on oeis.org

1, 1, 1, 4, 13, 146, 826, 44808, 1074557, 155741296, 10381741786, 6939251270348, 2203360264480750, 4186526735251514044, 3747344007864300197810, 35041787059621536192399824, 156277111373298355107598128061, 4142122641757597729416733678931968

Offset: 1

Gus Wiseman, Dec 22 2018

nonn

A graph is regular if all vertices have the same degree.

Gus Wiseman, The a(5) = 13 connected regular simple graphs.
Gus Wiseman, The a(6) = 146 connected regular simple graphs.

Row sums of A324163.

Cf. A054921, A059441, A062740, A295193, A299353, A305078, A319190, A319612, A319729, A322635.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[If[n==1,1,Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],SameQ@@Length/@Split[Sort[Join@@#]],Length[csm[#]]==1]&]]],{n,6}]

a(8)-a(15) from Andrew Howroyd, Dec 23 2018

a(16)-a(18) from Andrew Howroyd, Sep 02 2019

A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 21, 29, 4, 1, 0, 112, 2101, 150, 5, 1, 0, 853, 7011181, 7013164, 1037, 6, 1, 0, 11117, 1788775603301, 29281354507753847, 1788782615612, 12338, 7, 1, 0, 261080, 53304526022885278403, 234431745534048893449761040648508, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1

Offset: 1

Gus Wiseman, Jun 19 2018

			Triangle begins:
   1
   0    1
   0    2       1
   0    6       3       1
   0   21      29       4    1
   0  112    2101     150    5 1
   0  853 7011181 7013164 1037 6 1
   ...
The T(4,2) = 6 hypergraphs:
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Row sums are A301920.
Columns k=2..3 are A001349(n > 1), A003190(n > 1).
Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.

InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}
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}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}
A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}
{ my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 26 2019

Column k is the inverse Euler transform of column k of A301922. - Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

cp's OEIS Frontend

A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.

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A295193 Number of regular simple graphs on n labeled nodes.

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A319189 Number of uniform regular hypergraphs spanning n vertices.

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A322635 Number of regular graphs with loops on n labeled vertices.

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A322785 Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

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Mathematica

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A301920 Number of unlabeled uniform connected hypergraphs spanning n vertices.

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A299354 Regular triangle where T(n,k) is the number of labeled connected k-uniform hypergraphs spanning n vertices.

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Mathematica

Formula

A321698 MM-numbers of uniform regular multiset multisystems. Numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

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