A006649 -id:A006649 - cp's OEIS frontend

A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

1, 1, 4, 23, 193, 2133, 29410, 486602, 9395315, 207341153, 5147194204, 141939786588, 4304047703755, 142317774817901, 5095781837539766, 196403997108015332, 8106948166404074281, 356781439557643998591, 16675999433772328981216, 824952192369049982670686

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

These are covering loop-graphs where every connected component has a number of edges less than or equal to the number of vertices in that component. Also covering loop-graphs with at most one cycle (unicyclic) in each connected component.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{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}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

For a unique choice we have A000272, covering case of A088957.
Without the choice condition we have A322661, unlabeled A322700.
For exactly n edges we have A333331 (maybe), complement A368596.
The case without loops is A367869, covering case of A133686.
This is the covering case of A368927.
The complement is counted by A369142, covering case of A369141.
The unlabeled version is the first differences of A369145.
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.
A367862 counts graphs with n vertices and n edges, covering A367863.
Cf. A000169, A000666, A003465, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

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

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(-x + 3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ Andrew Howroyd, Feb 02 2024

Inverse binomial transform of A368927.
Exponential transform of A369197.
E.g.f.: exp(-x)*exp(3*T(x)/2 - 3*T(x)^2/4)/sqrt(1-T(x)), where T(x) is the e.g.f. of A000169. - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A369191 Number of labeled simple graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5686, 102084, 2162168, 52693975, 1450876804, 44509105965, 1504709144203, 55563209785167, 2224667253972242, 95984473918245388, 4439157388017620554, 219067678811211857307, 11489425098298623161164, 638159082104453330569185

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A054548.

			The a(0) = 1 through a(3) = 4 graphs:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}

The minimal case is A053530.
The connected case is A129271, unlabeled version A005703.
The case of equality is A367863, covering case of A367862.
This is the covering case of A369192, or A369193 for covered vertices.
The version for loop-graphs is A369194.
The unlabeled version is A370316.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A057500 counts connected graphs with n vertices and n edges.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001429, A003465, A006649, A143543, A116508, A140638, A322661.

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

Inverse binomial transform of A369193.

A369192 Number of labeled simple graphs with n vertices and at most n edges (not necessarily covering).

Original entry on oeis.org

1, 1, 2, 8, 57, 638, 9949, 198440, 4791323, 135142796, 4346814276, 156713948672, 6251579884084, 273172369790743, 12969420360339724, 664551587744173992, 36543412829258260135, 2146170890448154922648, 134053014635659737513358, 8872652968135849629240560

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The version for loop-graphs is A066383, covering A369194.
The case of equality is A116508, covering A367863, also A367862.
The connected case is A129271, unlabeled A005703.
The covering case is A369191, minimal case A053530.
Counting only covered vertices gives A369193.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A057500, A143543.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[#]<=n&]],{n,0,5}]

from math import comb
def A369192(n): return sum(comb(comb(n,2),k) for k in range(n+1)) # Chai Wah Wu, Jul 14 2024

a(n) = Sum_{k=0..n} binomial(binomial(n,2),k).

A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 0, 1, 4, 5, 5, 4, 2, 1, 1, 0, 0, 0, 1, 3, 9, 15, 20, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 1, 6, 20, 41, 73, 110, 133, 139, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 1, 3, 15, 50, 124, 271, 515, 832, 1181, 1460, 1581, 1516, 1291, 970, 658, 400, 220, 114, 56, 24, 11, 5, 2, 1, 1

Offset: 0

Gus Wiseman, Feb 15 2024

			Triangle begins:
  1
  0
  0  1
  0  0  1  1
  0  0  1  2  2  1  1
  0  0  0  1  4  5  5  4  2  1  1
  0  0  0  1  3  9 15 20 22 20 14  9  5  2  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)

Column sums are A000664.
Row sums are A002494.
This is the covering case of A008406, labeled A084546.
The labeled version is A054548, row sums A006129, column sums A121251.
The connected case is A054924, row sums A001349, column sums A002905.
The labeled connected case is A062734, with loops A369195.
The connected case with loops is A283755, row sums A054921.
The labeled version w/ loops is A369199, row sums A322661, col sums A173219.
Cf. A000666, A006125, A006649, A054547, A066383, A322700.

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],{2}],{k}],Union@@#==Range[n]&]]], {n,0,5},{k,0,Binomial[n,2]}]

\\ G(n) defined in A008406.
row(n)={Vecrev(G(n)-if(n>0, G(n-1)), binomial(n,2)+1)}
{ for(n=0, 7, print(row(n))) } \\ Andrew Howroyd, Feb 19 2024

a(42) onwards from Andrew Howroyd, Feb 19 2024

A054780 Number of n-covers of a labeled n-set.

Original entry on oeis.org

1, 1, 3, 32, 1225, 155106, 63602770, 85538516963, 386246934638991, 6001601072676524540, 327951891446717800997416, 64149416776011080449232990868, 45546527789182522411309599498741023, 118653450898277491435912500458608964207578

Offset: 0

Vladeta Jovovic, May 21 2000

Also, number of n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

			From _Gus Wiseman_, Dec 19 2023: (Start)
Number of ways to choose n nonempty sets with union {1..n}. For example, the a(3) = 32 covers are:
  {1}{2}{3}  {1}{2}{13}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{23}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{3}{12}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{23}  {1}{12}{23}  {2}{12}{123}
             {2}{3}{12}  {1}{13}{23}  {2}{13}{123}
             {2}{3}{13}  {2}{3}{123}  {2}{23}{123}
                         {2}{12}{13}  {3}{12}{123}
                         {2}{12}{23}  {3}{13}{123}
                         {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
(End)

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

Main diagonal of A055154.
Cf. A048291, A088310, A181230, A259763.
Covers with any number of edges are counted by A003465, unlabeled A055621.
Connected graphs of this type are counted by A057500, unlabeled A001429.
This is the covering case of A136556.
The case of graphs is A367863, covering case of A116508, unlabeled A006649.
Binomial transform is A367916.
These set-systems have ranks A367917.
The unlabeled version is A368186.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A046165 counts minimal covers, ranks A309326.
Cf. A006126, A058891, A059201, A133686, A323816, A323817, A323818, A326754, A367862, A367901.

Join[{1}, Table[Sum[StirlingS1[n+1, k+1]*(2^k - 1)^n, {k, 0, n}]/n!, {n, 1, 15}]] (* Vaclav Kotesovec, Jun 04 2022 *)
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]],{n}],Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

a(n) = sum(k=0, n, (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n)) \\ Andrew Howroyd, Jan 20 2024

a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*binomial(2^(n-k)-1, n).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n+1, k+1)*(2^k-1)^n.
G.f.: Sum_{n>=0} log(1+(2^n-1)*x)^n/((1+(2^n-1)*x)*n!). - Paul D. Hanna and Vladeta Jovovic, Jan 16 2008
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jun 04 2022
Inverse binomial transform of A367916. - Gus Wiseman, Dec 19 2023

A369146 Number of unlabeled loop-graphs with up to n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 8, 60, 471, 4911, 78797, 2207405, 113740613, 10926218807, 1956363413115, 652335084532025, 405402273420833338, 470568642161119515627, 1023063423471189429817807, 4178849203082023236054797465, 32168008290073542372004072630072, 468053896898117580623237189882068990

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

			The a(0) = 0 through a(3) = 8 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{1,2}}
                         {{1},{2},{3},{1,2}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

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

Without the choice condition we have A000666, labeled A006125 (shifted).
For a unique choice we have A087803, labeled A088957.
The case without loops is A140637, labeled A367867 (covering A367868).
For exactly n edges we have A368835, labeled A368596.
The labeled complement is A368927, covering A369140.
The labeled version is A369141, covering A369142.
The complement is counted by A369145, covering A369200.
The covering case is A369147.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
Cf. A000088, A000612, A006649, A062740, A134964, A137917, A140638, A368600, A368984, A369199.

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}]], Select[Tuples[#],UnsameQ@@#&]=={}&]]],{n,0,4}]

Partial sums of A369147.

a(n) = A000666(n) - A369145(n). - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 2, 5, 12, 30, 73, 185, 467, 1207, 3147, 8329, 22245, 60071, 163462, 448277, 1236913, 3432327, 9569352, 26792706, 75288346, 212249873, 600069431, 1700826842, 4831722294, 13754016792, 39224295915, 112048279650, 320563736148, 918388655873, 2634460759783, 7566000947867

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

a(n) is the number of graphs with loops on n unlabeled vertices with every connected component having no more edges than vertices. - Andrew Howroyd, Feb 02 2024

			The a(0) = 1 through a(3) = 12 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{1},{2}}    {{1},{2}}
             {{1},{1,2}}  {{1},{1,2}}
                          {{1},{2,3}}
                          {{1,2},{1,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}

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

Without the choice condition we get A000666, labeled A006125 (shifted left).
The case of a unique choice is A087803, labeled A088957.
Without loops we have A134964, labeled A133686 (covering A367869).
For exactly n edges and no loops we have A137917, labeled A137916.
The labeled version is A368927, covering A369140.
The labeled complement is A369141, covering A369142.
For exactly n edges we have A368984, labeled A333331 (maybe).
The complement for exactly n edges is A368835, labeled A368596.
The complement is counted by A369146, labeled A369141 (covering A369142).
The covering case is A369200.
The complement for exactly n edges and no loops is A369201, labeled A369143.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable labeled graphs, covering A367868.
A368927 counts choosable labeled loop-graphs, covering A369140.
Cf. A000088, A006649, A062740, A140638, A368924, A369194.

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}]], Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

Partial sums of A369200.

Euler transform of A369289. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. 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.

			The terms together with the corresponding set-systems begin:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

These set-systems are counted by A054780 and A367916, A368186.
Graphs of this type are A367862, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, connected A323818, unlabeled A000612.
A070939 gives length of binary expansion.
A136556 counts set-systems on {1..n} with n edges.
Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Select[Range[0,100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

Original entry on oeis.org

1, 2, 7, 39, 320, 3584, 51405, 900947, 18661186, 445827942, 12062839691, 364451604095, 12157649050827, 443713171974080, 17583351295466338, 751745326170662049, 34485624653535808340, 1689485711682987916502, 88030098291829749593643, 4860631073631586486397141

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(2) = 7 loop-graphs:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

The version counting all vertices is A066383, without loops A369192.
The loopless case is A369193, with case of equality A367862.
The covering case is A369194, connected A369197, minimal case A001862.
The case of equality is A369198, covering case A368597.
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. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

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

Binomial transform of A369194.

A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

Original entry on oeis.org

1, 1, 2, 8, 57, 608, 8614, 151365, 3162353, 76359554, 2088663444, 63760182536, 2147325661180, 79051734050283, 3157246719905273, 135938652662043977, 6275929675565965599, 309242148569525451140, 16197470691388774460758, 898619766673014862321176, 52639402023471657682257626

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The case of equality is A367862, covering case of A116508, also A367863.
The covering case is A369191, for loop-graphs A369194.
The version counting all vertices is A369192.
The version for loop-graphs is A369196, counting all vertices A066383.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A053530, A129271, A143543, A322661, A367916.

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

Binomial transform of A369191.

cp's OEIS Frontend

A369140 Number of labeled loop-graphs covering {1..n} such that it is possible to choose a different vertex from each edge (choosable).

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A369191 Number of labeled simple graphs covering n vertices with at most n edges.

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A369192 Number of labeled simple graphs with n vertices and at most n edges (not necessarily covering).

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A370167 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with k = 0..binomial(n,2) edges.

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Mathematica

PARI

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A054780 Number of n-covers of a labeled n-set.

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A369146 Number of unlabeled loop-graphs with up to n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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Mathematica

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A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

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