A368836 -id:A368836 - cp's OEIS frontend

A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.

1, 1, 2, 5, 12, 29, 75, 191, 504, 1339, 3610, 9800, 26881, 74118, 205706, 573514, 1606107, 4513830, 12727944, 35989960, 102026638, 289877828, 825273050, 2353794251, 6724468631, 19239746730, 55123700591, 158133959239, 454168562921, 1305796834570, 3758088009136

Offset: 0

Andrew Howroyd, Jan 11 2024

nonn

The graphs considered here can have loops but not parallel edges.

Also the number of unlabeled loop-graphs with n edges and n vertices such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

			Representatives of the a(3) = 5 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}},
   {{1}, {2}, {2,3}},
   {{1}, {2}, {3}}.
The graph with 4 vertices and edges {{1}, {2}, {1,2}, {3,4}} is included by A368599 but not by this sequence.

Andrew Howroyd, Table of n, a(n) for n = 0..500

The case of a unique choice is A000081.
Without loops we have A137917, labeled A137916.
The labeled version appears to be A333331.
Without the choice condition we have A368598, covering A368599.
The complement is counted by A368835, labeled A368596 (covering A368730).
Row sums of A368926, labeled A368924.
The connected case is A368983.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A014068, A057500, A116508, A129271, A133686, A367863, A367869, A367902, A368597, A368601, A368836.

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}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

Euler transform of A368983.

A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 173, 585, 2064, 7520, 28265, 109501, 437394, 1799843, 7629463, 33302834, 149633151, 691702799, 3287804961, 16058229900, 80533510224, 414384339438, 2185878202630, 11811050484851, 65318772618624, 369428031895444, 2135166786135671, 12601624505404858

Offset: 0

Gus Wiseman, Jan 05 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}        {{1},{2},{3},{4}}
             {{1},{1,2}}  {{1},{2},{1,2}}      {{1},{2},{3},{1,2}}
                          {{1},{2},{1,3}}      {{1},{2},{3},{1,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}    {{1},{2},{1,2},{3,4}}
                          {{1,2},{1,3},{2,3}}  {{1},{2},{1,3},{1,4}}
                                               {{1},{2},{1,3},{2,3}}
                                               {{1},{2},{1,3},{2,4}}
                                               {{1},{3},{1,2},{2,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,3}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{2},{1,2},{1,3},{1,4}}
                                               {{4},{1,2},{1,3},{2,3}}
                                               {{1,2},{1,3},{1,4},{2,3}}
                                               {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.

For any number of edges of any size we have A000612, covering A055621.
For any number of edges we have A000666, A054921, A322700.
The labeled version is A014068.
Counting by weight gives A320663, or A339888 with loops {x,x}.
The covering case is A368599.
For edges of any size we have A368731, covering A368186.
Row sums of A368836.
A000085 counts set partitions into singletons or pairs.
A001515 counts length-n set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A001434, A007716, A007717, A058891, A070166, A122848, A124059, A283877, A302545, A322661, A339741, A339887, A370168.

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 /@ Subsets[Subsets[Range[n],{1,2}],{n}]]],{n,0,5}]

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

a(n) = A070166(n, n). - Andrew Howroyd, Jan 09 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 09 2024

A368835 Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

Original entry on oeis.org

0, 0, 0, 1, 5, 23, 98, 394, 1560, 6181, 24655, 99701, 410513, 1725725, 7423757, 32729320, 148027044, 687188969, 3275077017, 16022239940, 80431483586, 414094461610, 2185052929580, 11808696690600, 65312048149993, 369408792148714, 2135111662435080, 12601466371445619

Offset: 0

Gus Wiseman, Jan 13 2024

nonn

			Non-isomorphic representatives of the a(4) = 5 loop-graphs:
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{1,2},{1,3}}
  {{1,1},{2,2},{1,2},{3,4}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{2,3}}

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

The case of a unique choice is A000081, row sums of A106234.
The labeled version is A368596, covering A368730.
Without the choice condition we have A368598.
The complement is A368984, row sums of A368926.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A116508, A129271, A133686, A137916, A137917, A333331, A367863, A368836, A368924, A368927.

Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,5}]

a(n) = A368598(n) - A368984(n). - Andrew Howroyd, Jan 14 2024

a(8) onwards from Andrew Howroyd, Jan 14 2024

A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 9, 6, 1, 15, 68, 48, 12, 1, 222, 720, 510, 150, 20, 1, 3670, 9738, 6825, 2180, 360, 30, 1, 68820, 159628, 110334, 36960, 6895, 735, 42, 1, 1456875, 3067320, 2090760, 721560, 145530, 17976, 1344, 56, 1, 34506640, 67512798, 45422928, 15989232, 3402756, 463680, 40908, 2268, 72, 1

Offset: 0

Gus Wiseman, Jan 10 2024

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.

			Triangle begins:
      1
      0      1
      0      2      1
      1      9      6      1
     15     68     48     12      1
    222    720    510    150     20      1
   3670   9738   6825   2180    360     30      1
  68820 159628 110334  36960   6895    735     42      1
Row n = 3 counts the following loop-graphs:
  {{1,2},{1,3},{2,3}}  {{1},{1,2},{1,3}}  {{1},{2},{1,3}}  {{1},{2},{3}}
                       {{1},{1,2},{2,3}}  {{1},{2},{2,3}}
                       {{1},{1,3},{2,3}}  {{1},{3},{1,2}}
                       {{2},{1,2},{1,3}}  {{1},{3},{2,3}}
                       {{2},{1,2},{2,3}}  {{2},{3},{1,2}}
                       {{2},{1,3},{2,3}}  {{2},{3},{1,3}}
                       {{3},{1,2},{1,3}}
                       {{3},{1,2},{2,3}}
                       {{3},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Axiom of choice.

Column k = n-1 is A002378.
The case of a unique choice is A061356, row sums A000272.
Column k = 0 is A137916, unlabeled version A137917.
Row sums appear to be A333331.
The complement has row sums A368596, covering case A368730.
The unlabeled version is A368926.
Without the choice condition we have A368928, A116508, A367863, A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
Cf. A000169, A057500, A062740, A129271, A133686, A322661, A367869, A367902, A368601, A368835, A368836, A368927.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]],{n,0,5},{k,0,n}]

T(n)={my(t=-lambertw(-x + O(x*x^n))); [Vecrev(p) | p <- Vec(serlaplace(exp(-log(1-t)/2 - t/2 + t*y - t^2/4)))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 14 2024

E.g.f.: A(x,y) = exp(-log(1-T(x))/2 - T(x)/2 + y*T(x) - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 14 2024

a(36) onwards from Andrew Howroyd, Jan 14 2024

A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 5, 3, 1, 1, 5, 12, 7, 3, 1, 1, 14, 29, 19, 8, 3, 1, 1, 35, 75, 47, 21, 8, 3, 1, 1, 97, 191, 127, 54, 22, 8, 3, 1, 1, 264, 504, 331, 149, 56, 22, 8, 3, 1, 1, 733, 1339, 895, 395, 156, 57, 22, 8, 3, 1, 1

Offset: 0

Gus Wiseman, Jan 13 2024

Also the number of unlabeled loop-graphs covering n vertices with k loops and n-k non-loops such that each connected component has the same number of edges as vertices.

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  1  1
   2  5  3  1  1
   5 12  7  3  1  1
  14 29 19  8  3  1  1
  35 75 47 21  8  3  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

The case of a unique choice is A106234, row sums A000081.
Column k = 0 is A137917, labeled version A137916.
Without the choice condition we have A368836.
The labeled version is A368924, row sums maybe A333331.
Row sums are A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.

Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]], {n,0,5},{k,0,n}]

\\ TreeGf gives gf of A000081; G(n,1) is gf of A368983.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
G(n,y)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); 1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2 + (y-1)*g(1)}
EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n,y) - 1))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2024

a(36) onwards from Andrew Howroyd, Jan 14 2024

A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 9, 9, 1, 15, 80, 90, 24, 1, 252, 1050, 1200, 450, 50, 1, 5005, 18018, 20475, 9100, 1575, 90, 1, 116280, 379848, 427329, 209475, 46550, 4410, 147, 1, 3108105, 9472320, 10548720, 5503680, 1433250, 183456, 10584, 224, 1

Offset: 0

Gus Wiseman, Jan 11 2024

			Triangle begins:
     1
     0     1
     0     2     1
     1     9     9     1
    15    80    90    24     1
   252  1050  1200   450    50     1
  5005 18018 20475  9100  1575    90     1
The loop-graphs counted in row n = 3 (loops shown as singletons):
  {12}{13}{23}  {1}{12}{13}  {1}{2}{12}  {1}{2}{3}
                {1}{12}{23}  {1}{2}{13}
                {1}{13}{23}  {1}{2}{23}
                {2}{12}{13}  {1}{3}{12}
                {2}{12}{23}  {1}{3}{13}
                {2}{13}{23}  {1}{3}{23}
                {3}{12}{13}  {2}{3}{12}
                {3}{12}{23}  {2}{3}{13}
                {3}{13}{23}  {2}{3}{23}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A014068, unlabeled version A000666.
Column k = 0 is A116508, covering version A367863.
The covering case is A368597.
The unlabeled version is A368836.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A057500, A079491, A339065, A368596, A368927.

Table[Length[Select[Subsets[Subsets[Range[n], {1,2}],{n}],Count[#,{_}]==k&]],{n,0,5},{k,0,n}]
T[n_,k_]:= Binomial[n,k]*Binomial[Binomial[n,2],n-k]; Table[T[n,k],{n,0,8},{k,0,n}]// Flatten (* Stefano Spezia, Jan 14 2024 *)

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k) \\ Andrew Howroyd, Jan 14 2024

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k).

cp's OEIS Frontend

A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A368835 Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A368924 Triangle read by rows where T(n,k) is the number of labeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different vertex from each edge.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula