A005176 -id:A005176 - cp's OEIS frontend

A185318 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 8.

1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 9, 8, 11, 11, 14, 14, 19, 18, 23, 24, 30, 31, 41, 40, 61, 52, 217, 67, 4416, 86, 266463, 111, 20807816, 141

Offset: 0

Jason Kimberley, Dec 18 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), this sequence (g=8), A185319 (g=9).

a(n) = A186728(n) + A185218(n).

A185319 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 9.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 8, 10, 10, 13, 13, 16, 17, 20, 21, 26, 27, 32, 35, 41, 44, 52, 56, 65, 72, 82, 90, 104, 114, 130, 144, 163, 180, 205, 226, 255, 283, 336

Offset: 0

Jason Kimberley, Dec 19 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected regular simple graphs with girth at least g: A005176 (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), this sequence (g=9).

a(n) = A186729(n) + A185219(n).

A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1

Offset: 1

Gus Wiseman, Dec 17 2018

			Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Andrew Howroyd, Table of n, a(n) for n = 1..300

Row sums are A322555.

Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

A322784 Number of 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, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120

Offset: 0

Gus Wiseman, Dec 26 2018

nonn

A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020

			The a(1) = 1 through a(4) = 29 multiset partitions:
  {{1}}   {{1,1}}     {{1,1,1}}       {{1,1,1,1}}
          {{1,2}}     {{1,2,3}}       {{1,1,2,2}}
         {{1},{1}}   {{1},{1,1}}      {{1,2,3,4}}
         {{1},{2}}   {{1},{2,3}}     {{1},{1,1,1}}
                     {{2},{1,3}}     {{1,1},{1,1}}
                     {{3},{1,2}}     {{1},{1,2,2}}
                    {{1},{1},{1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}    {{1,2},{1,2}}
                                     {{1},{2,3,4}}
                                     {{1,2},{3,4}}
                                     {{1,3},{2,4}}
                                     {{1,4},{2,3}}
                                     {{2},{1,1,2}}
                                     {{2},{1,3,4}}
                                     {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                    {{1},{1},{1,1}}
                                    {{1},{1},{2,2}}
                                    {{1},{2},{1,2}}
                                    {{1},{2},{3,4}}
                                    {{1},{3},{2,4}}
                                    {{1},{4},{2,3}}
                                    {{2},{2},{1,1}}
                                    {{2},{3},{1,4}}
                                    {{2},{4},{1,3}}
                                    {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{3},{4}}

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

Row sums of A322786.

Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.

u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Sum[u[Array[Prime,d,1,Times]^(n/d),Array[Prime,d,1,Times]^(n/d)],{d,Divisors[n]}],{n,12}]

a(n) = Sum_{d|n} A001055(A002110(n/d)^d).

a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020

a(14)-a(15) from Alois P. Heinz, Jan 16 2019

Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020

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[#]]]&]

A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 2, 2, 5, 4, 3, 2, 2, 27, 11, 6, 4, 2, 2, 142, 29, 8, 4, 282, 12, 3, 1073, 101, 8, 4, 2, 2, 32034, 1581, 234, 75, 20, 6, 2, 2, 136853, 2660, 10, 4, 1527528, 1985, 91, 4, 4661087, 64596, 648, 20, 5, 2, 2, 227932993, 1280333, 41945, 231, 28, 6

Offset: 1

Gus Wiseman, Dec 26 2018

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

			Triangle begins:
     1
     2    2
     2    2
     5    4    3
     2    2
    27   11    6    4
     2    2
   142   29    8    4
   282   12    3
  1073  101    8    4
The multiset partitions counted under row 6:
  {123456}          {112233}          {111222}          {111111}
  {123}{456}        {112}{233}        {111}{222}        {111}{111}
  {124}{356}        {113}{223}        {112}{122}        {11}{11}{11}
  {125}{346}        {122}{133}        {11}{12}{22}      {1}{1}{1}{1}{1}{1}
  {126}{345}        {123}{123}        {12}{12}{12}
  {134}{256}        {11}{22}{33}      {1}{1}{1}{2}{2}{2}
  {135}{246}        {11}{23}{23}
  {136}{245}        {12}{12}{33}
  {145}{236}        {12}{13}{23}
  {146}{235}        {13}{13}{22}
  {156}{234}        {1}{1}{2}{2}{3}{3}
  {12}{34}{56}
  {12}{35}{46}
  {12}{36}{45}
  {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}{2}{3}{4}{5}{6}

Andrew Howroyd, Table of n, a(n) for n = 1..482 (rows 1..100)

Row sums are A322785. First column is A038041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.

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[Length[Select[mps[Join@@Table[Range[n/d],{d}]],SameQ@@Length/@#&]],{n,10},{d,Divisors[n]}]

T(n,k) = A322794(A002110(n/d)^d), where d = A027750(n,k).

More terms from Alois P. Heinz, Jan 30 2019
Terms a(38) and beyond from Andrew Howroyd, Feb 03 2022
Edited by Peter Munn, Mar 05 2025

A350913 Number of oriented graphs on n unlabeled nodes whose underlying graph is regular.

Original entry on oeis.org

1, 1, 2, 3, 10, 17, 244, 2077, 181018, 12150445, 6986854715, 2456586421099, 12983287104044905, 40482495806423235417, 1213455558602787270259889, 29044259315697996254338202296, 6058969047090883735962137093497249, 785177832305082826579135707562767426031

Offset: 0

Andrew Howroyd, Jan 29 2022

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..22

The labeled version is A351264.
Row sums of A350912.
Cf. A001174, A005176, A350911.

A076434 Number of weakly regular simple graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 4, 16, 21, 168, 544, 18992, 389451, 50314864

Offset: 1

Eric W. Weisstein, Oct 11 2002

Eric Weisstein's World of Mathematics, Weakly Regular Graph

Cf. A005176 (regular simple graphs on n nodes).
Cf. A294405 (connected weakly regular graphs on n nodes).
Cf. A076435 (strongly regular simple graphs on n nodes).

a(n) = A005176(n) - A076435(n). - Andrew Howroyd, Sep 03 2019

a(10) from Eric W. Weisstein, Oct 30 2017

a(11)-a(14) from Sean A. Irvine, Apr 04 2025

A076435 Number of strongly regular simple graphs on n nodes.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 6, 5, 8, 2, 10, 3, 6

Offset: 1

Eric W. Weisstein, Oct 11 2002

			1: K_1 (1 graph)
2: \bar K_2, P_2 = K_2 (2 graph)
3: \bar K_3, C_3 = K_3 (2 graphs)
4: \bar K_4, 2P_2, C_4, K_4 (4 graphs)
5: C_5, \bar K_5, K_5 (3 graphs)
6: 2C_3, K_6, \bar K_6, 3P_2, Ci_6(1,2), K_{3,3} (6 graphs)
Here, \bar denotes the graph complement and Ci_n(...) a circular graph.

Eric Weisstein's World of Mathematics, Strongly Regular Graph

Cf. A005176, A076434, A088741.

a(10) from Eric W. Weisstein, Oct 30 2017

a(11)-a(14) from Sean A. Irvine, Sep 11 2021

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

A185318 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 8.

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A185319 Number of, not necessarily connected, regular simple graphs on n vertices with girth at least 9.

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A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

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A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

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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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A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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A350913 Number of oriented graphs on n unlabeled nodes whose underlying graph is regular.

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A076434 Number of weakly regular simple graphs on n nodes.

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A076435 Number of strongly regular simple graphs on n nodes.

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