A319189 -id:A319189 - cp's OEIS frontend

A295193 Number of regular simple graphs on n labeled nodes.

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

A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1

Offset: 1

N. J. A. Sloane, Feb 01 2001

			1;
1,   1;
1,   0,       1;
1,   3,       3,        1;
1,   0,      12,        0,          1;
1,  15,      70,       70,         15,    1;
1,   0,     465,        0,        465,    0,   1;
1, 105,    3507,    19355,      19355, 3507, 105, 1;
1,   0,   30016,        0,    1024380, ...;
1, 945,  286884, 11180820,   66462606, ...;
1,   0, 3026655,        0, 5188453830, ...;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.
Wikipedia, Regular graph

Row sums are A295193.
Columns: A123023 (k=1), A001205 (k=2), A002829 (k=3, with alternating zeros), A005815 (k=4), A338978 (k=5, with alternating zeros), A339847 (k=6).
Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).
Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.

Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{n,9},{k,0,n-1}] (* Gus Wiseman, Dec 24 2018 *)

for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(37)-a(55) from Andrew Howroyd, Aug 25 2017

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

A319612 Number of regular simple graphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 1, 7, 13, 171, 931, 45935, 1084413, 155862511, 10382960971, 6939278572095, 2203360500122299, 4186526756621772343, 3747344008241368443819, 35041787059691023579970847, 156277111373303386104606663421, 4142122641757598618318165240180095

Offset: 0

Gus Wiseman, Dec 17 2018

nonn

A graph is regular if all vertices have the same degree. The span of a graph is the union of its edges.

			The a(4) = 7 edge-sets:
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{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},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A002829, A005176, A110100, A116539, A295193, A306017, A319189, A319190.

a(n) = A295193(n) - 1.

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

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

A322698 Number of regular graphs with half-edges on n labeled vertices.

Original entry on oeis.org

1, 2, 4, 10, 40, 278, 3554, 84590, 3776280, 317806466, 50710452574, 15414839551538, 8964708979273634, 10008446308186072290, 21518891146915893435358, 89320970210116481106835986, 717558285660687970023516336792, 11176382741327158622885664697124082, 338202509574712032788035618665293979610

Offset: 0

Gus Wiseman, Dec 23 2018

nonn

A graph is regular if all vertices have the same degree. A half-edge is like a loop except it only adds 1 to the degree of its vertex.

			The a(3) = 10 edge sets:
  {}
  {{1},{2,3}}
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Wikipedia, Regular graph

Row sums of A333157.

Cf. A058891, A059441, A116539, A283877, A295193, A319189, A319190, A319612, A319729.

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

for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(10)-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

A326785 BII-numbers of uniform regular set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 16, 32, 52, 64, 128, 129, 130, 131, 136, 137, 138, 139, 256, 288, 512, 528, 772, 816, 1024, 2048, 2052, 2320, 2340, 2580, 2592, 2868, 4096, 8192, 13376, 16384, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897

Offset: 1

Gus Wiseman, Jul 25 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges. A set-system is uniform if all edges have the same size, and regular if all vertices appear the same number of times.

			The sequence of all uniform regular set-systems together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   16: {{1,3}}
   32: {{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
  130: {{2},{4}}
  131: {{1},{2},{4}}
  136: {{3},{4}}
  137: {{1},{3},{4}}
  138: {{2},{3},{4}}

Cf. A000120, A029931, A048793, A070939, A319056, A319189, A321698, A326031, A326701, A326783 (uniform), A326784 (regular), A326788.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],SameQ@@Length/@bpe/@bpe[#]&&SameQ@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]&]

Intersection of A326783 and A326784.

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[#]]]&]

A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 49, 53, 64, 81, 113, 128, 131, 151, 161, 165, 169, 225, 243, 256, 311, 343, 361, 512, 719, 729, 1024, 1291, 1321, 1619, 1937, 1957, 2021, 2048, 2093, 2117, 2187, 2197, 2257, 2401, 2805, 2809, 3375, 3671, 4096, 6561

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, regular, and spans an initial interval of positive integers, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
    1: {}
    2: {{}}
    3: {{1}}
    4: {{},{}}
    7: {{1,1}}
    8: {{},{},{}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   16: {{},{},{},{}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   32: {{},{},{},{},{}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   64: {{},{},{},{},{},{}}
   81: {{1},{1},{1},{1}}
  113: {{1,2,3}}
  128: {{},{},{},{},{},{},{}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  165: {{1},{2},{3}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}
  243: {{1},{1},{1},{1},{1}}
  256: {{},{},{},{},{},{},{},{}}
  311: {{1,1,1,1,1,1}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}
  512: {{},{},{},{},{},{},{},{},{}}
  719: {{1,1,1,1,1,1,1}}
  729: {{1},{1},{1},{1},{1},{1}}

Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

cp's OEIS Frontend

A295193 Number of regular simple graphs on n labeled nodes.

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A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

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A319190 Number of regular hypergraphs spanning n vertices.

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A319612 Number of regular simple graphs spanning n vertices.

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

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A322698 Number of regular graphs with half-edges 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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A326785 BII-numbers of uniform regular set-systems.

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