cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-36 of 36 results.

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 19 2024

Keywords

Comments

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

Examples

			Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

Links

Crossrefs

Row lengths are A000124.
Diagonal T(n,n-1) is A000272, rooted A000169.
The case without loops is A062734.
Row sums are A062740.
Transpose is A322147.
Column sums are A322151.
Diagonal T(n,n) is A368951, connected case of A368597.
Connected case of A369199, without loops A054548.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]
  • PARI
    T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

Formula

E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

Original entry on oeis.org

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560
Offset: 0

Views

Author

Gus Wiseman, Jan 06 2024

Keywords

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Links

Crossrefs

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

Programs

  • Mathematica
    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]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
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], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Formula

Inverse Euler transform of A320663.

A368727 Number of non-isomorphic connected multiset partitions of weight n into singletons or strict pairs.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 15, 21, 49, 82, 184, 341, 766, 1530, 3428, 7249, 16394, 36009, 82492, 186485, 433096, 1001495, 2358182, 5554644, 13255532, 31718030, 76656602, 185982207, 454889643, 1117496012, 2764222322, 6868902152, 17172601190
Offset: 0

Views

Author

Gus Wiseman, Jan 06 2024

Keywords

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 15 multiset partitions:
  {1}  {12}    {2}{12}    {12}{12}      {2}{12}{12}      {12}{12}{12}
       {1}{1}  {1}{1}{1}  {13}{23}      {2}{13}{23}      {12}{13}{23}
                          {1}{2}{12}    {3}{13}{23}      {13}{23}{23}
                          {2}{2}{12}    {1}{2}{2}{12}    {13}{24}{34}
                          {1}{1}{1}{1}  {2}{2}{2}{12}    {14}{24}{34}
                                        {1}{1}{1}{1}{1}  {1}{2}{12}{12}
                                                         {1}{2}{13}{23}
                                                         {2}{2}{12}{12}
                                                         {2}{2}{13}{23}
                                                         {2}{3}{13}{23}
                                                         {3}{3}{13}{23}
                                                         {1}{1}{2}{2}{12}
                                                         {1}{2}{2}{2}{12}
                                                         {2}{2}{2}{2}{12}
                                                         {1}{1}{1}{1}{1}{1}

Links

Crossrefs

For edges of any size we have A056156, with loops A007718.
This is the connected case of A339888.
Allowing loops {x,x} gives A368726, Euler transform A320663.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368598, A368599, A368731.

Programs

  • Mathematica
    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]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List /@ c[[1]]],Union@@s[[c[[1]]]]]]]]];
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],And@@UnsameQ@@@#&&Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Formula

Inverse Euler transform of A339888.

A370165 Number of labeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

Original entry on oeis.org

1, 1, 4, 29, 400, 10289, 496548, 45455677, 7983420736, 2716094133313, 1803251169342820, 2348787270663723581, 6024912118926389490448, 30516957491540079828757553, 305811332460677494410532494660, 6071677788061208810793717466942237
Offset: 0

Views

Author

Gus Wiseman, Feb 12 2024

Keywords

Comments

Number of ways to choose a stable vertex set of a simple graph with n vertices.

Examples

			The a(3) = 29 loop-graphs (loops shown as singletons):
  {1,23}   {1,2,3}     {1,2,13,23}
  {2,13}   {1,2,13}    {1,3,12,23}
  {3,12}   {1,2,23}    {2,3,12,13}
  {12,13}  {1,3,12}    {1,12,13,23}
  {12,23}  {1,3,23}    {2,12,13,23}
  {13,23}  {2,3,12}    {3,12,13,23}
           {2,3,13}
           {1,12,13}
           {1,12,23}
           {1,13,23}
           {2,12,13}
           {2,12,23}
           {2,13,23}
           {3,12,13}
           {3,12,23}
           {3,13,23}
           {12,13,23}

Links

Crossrefs

Without loops we have A006129, connected A001187.
The non-covering version is A079491.
The unlabeled version is A370166, non-covering A339832.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts labeled loop-graphs (shifted left), covering A322661.
Cf. A048291, A062740, A116539, A320461, A322635.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&!MatchQ[#, {_,{x_},_,{y_},_,{x_,y_},_}]&]],{n,0,5}]
  • PARI
    seq(n)={Vec(serlaplace(sum(k=0, n, exp((2^k-1)*x + O(x*x^n))*2^(k*(k-1)/2)*x^k/k!)))} \\ Andrew Howroyd, Feb 20 2024

Formula

Inverse binomial transform of A079491.
E.g.f.: Sum_{k >= 0} exp((2^k-1)*x)*2^(k*(k-1)/2)*x^k/k!. - Andrew Howroyd, Feb 20 2024

A369198 Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 30, 241, 2759, 40824, 736342, 15622835, 380668095, 10467815086, 320529284621, 10813165015074, 398413594789777, 15917197015926392, 685312404706694574, 31631317971844128229, 1558017329350990780607, 81567807853701988869120, 4522975947689168088308305
Offset: 0

Views

Author

Gus Wiseman, Jan 18 2024

Keywords

Examples

			The a(0) = 1 through a(3) = 30 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{2}}    {{3}}
             {{1},{1,2}}  {{1},{2}}
             {{2},{1,2}}  {{1},{3}}
                          {{2},{3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Crossrefs

The version counting all vertices is A014068.
The loopless case is A367862, counting all vertices A116508.
The covering case is A368597, connected A368951.
With inequality we have A369196, covering A369194, connected A369197.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000272, A000666, A006649, A062740, A066383, A367863, A369192, A369193.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]

Formula

Binomial transform of A368597.

A370166 Number of unlabeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

Original entry on oeis.org

1, 1, 3, 9, 36, 180, 1313, 14709, 277755, 9304977, 568315345, 63806703305, 13200565313255, 5042653259803433, 3567050969262370941, 4688444463558713135201, 11491940559865490367844649, 52719458629883487816297211441, 454220675869975957947658748125099
Offset: 0

Views

Author

Gus Wiseman, Feb 12 2024

Keywords

Examples

			Representatives of the a(0) = 1 through a(3) = 9 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{1,2},{1,3}}
             {{1},{1,2}}  {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}
                          {{1},{2},{1,3},{2,3}}
                          {{1},{1,2},{1,3},{2,3}}

Crossrefs

Without loops we have A002494, labeled A006129, connected A001349.
The non-covering version is A339832.
The labeled version is A370165, non-covering A079491 (apparently).
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts labeled loop-graphs (shifted left), covering A322661.
Cf. A000085, A001187, A048291, A062740, A116539.

Programs

  • Mathematica
    brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n] && !MatchQ[#,{_,{x_},_,{y_},_,{x_,y_},_}]&]]], {n,0,4}]

Formula

First differences of A339832 (the non-covering version).
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