A134964 -id:A134964 - cp's OEIS frontend

A133686 Number of labeled n-node graphs with at most one cycle in each connected component.

1, 1, 2, 8, 57, 608, 8524, 145800, 2918123, 66617234, 1704913434, 48300128696, 1499864341015, 50648006463048, 1847622972848648, 72406232075624192, 3033607843748296089, 135313823447621913500, 6402077421524339766058, 320237988317922139148736

Offset: 0

Washington Bomfim, May 12 2008

The total number of those graphs of order 5 is 608. The number of forests of trees on n labeled nodes of order 5 is 291, so the majority of the graphs of that kind have one or more unicycles.

Also the number of labeled graphs with n vertices satisfying a strict version of the axiom of choice. 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. The connected case is A129271, complement A140638. The unlabeled version is A134964. - Gus Wiseman, Dec 22 2023

			Below we see the 7 partitions of n=5 in the form c_1 + 2c_2 + ... + nc_n followed by the corresponding number of graphs. We consider the values of A129271(j) given by the table
   j|1|2|3| 4|  5|
----+-+-+-+--+---+
a(j)|1|1|4|31|347|
1*5 -> 5!1^5 / (1!^5 * 5!) = 1
2*1 + 1*3 -> 5!1^1 * 1^3 / (2!^1 * 1! * 1!^3 * 3!) = 10
2*2 + 1*1 -> 5!1^2 * 1^1 / (2!^2 * 2! * 1!^1 * 1!) = 15
3*1 + 1*2 -> 5!4^1 * 1^2 / (3!^1 * 1! * 1!^2 * 2!) = 40
3*1 + 2*1 -> 5!4^1 * 1^1 / (3!^1 * 1! * 2!^1 * 1!) = 40
4*1 + 1*1 -> 5!31^1 * 1^1 / (4!^1 * 1! * 1!^1 * 1!) = 155
5*1 -> 5!347^1 / (5!^1 * 1!) = 347
Total 608

Alois P. Heinz, Table of n, a(n) for n = 0..386
Wikipedia, Pseudoforest

Cf. A129137, A005703, A000272, A057500, A137916, A001858.
Row sums of triangle A144228. - Alois P. Heinz, Sep 15 2008
Cf. A137352. - Vladeta Jovovic, Sep 16 2008
The unlabeled version is A134964.
The complement is counted by A367867, covering A367868, connected A140638.
The covering case is A367869, connected A129271.
For set-systems we have A367902, ranks A367906.
The complement for set-systems is A367903, ranks A367907.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A143543 counts graphs by number of connected components.
Cf. A001187, A058891, A116508, A355740, A367769, A367770, A367863, A367901.

cy:= proc(n) option remember; binomial(n-1, 2)*
        add((n-3)!/(n-2-t)! *n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember;
      if k=0 then 1
    elif k<0 or n add(T(n,k), k=0..n):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 15 2008

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[ Exp[t/2-3t^2/4]/(1-t)^(1/2),{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)

x='x+O('x^50); Vec(serlaplace(sqrt(-lambertw(-x)/(x*(1+ lambertw(-x))))*exp(-(3/4)*lambertw(-x)^2))) \\ G. C. Greubel, Nov 16 2017

a(0) = 1; for n >=1, a(n) = Sum of n!prod_{j=1}^n\{ frac{ A129271(j)^{c_j} } { j!^{c_j}c_j! } } over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0.
a(n) = Sum_{k=0..n} A144228(n,k). - Alois P. Heinz, Sep 15 2008
E.g.f.: sqrt(-LambertW(-x)/(x*(1+LambertW(-x))))*exp(-3/4 * LambertW(-x)^2). - Vladeta Jovovic, Sep 16 2008
E.g.f.: A(x)*B(x) where A(x) is the e.g.f. for A137916 and B(x) is the e.g.f. for A001858. - Geoffrey Critzer, Mar 23 2013
a(n) ~ 2^(-1/4) * Gamma(3/4) * exp(-1/4) * n^(n-1/4) / sqrt(Pi) * (1-7*Pi/(12*Gamma(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Oct 08 2013
E.g.f.: exp(B(x) - 1) where B(x) is the e.g.f. of A129271. - Andrew Howroyd, Dec 30 2023

Corrected and extended by Alois P. Heinz and Vladeta Jovovic, Sep 15 2008

A129271 Number of labeled n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 4, 31, 347, 4956, 85102, 1698712, 38562309, 980107840, 27559801736, 849285938304, 28459975589311, 1030366840792576, 40079074477640850, 1666985134587145216, 73827334760713500233, 3468746291121007607808, 172335499299097826575564, 9027150377126199463936000

Offset: 0

Washington Bomfim, May 10 2008

The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.

Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024

			a(4) = 16 + 3*3 = 31.
From _Gus Wiseman_, Feb 19 2024: (Start)
The a(0) = 1 through a(3) = 4 graph edge sets:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, PseudoForest.

Cf. A129137, A000272.
For any number of edges we have A001187, unlabeled A001349.
The unlabeled version is A005703.
The case of equality is A057500, covering A370317, cf. A370318.
The non-connected non-covering version is A133686.
The connected complement is A140638, unlabeled A140636, covering A367868.
The non-connected covering version is A367869 or A369191.
The version with loops is A369197, non-connected A369194.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by number of edges.
Cf. A006649, A116508, A134964, A143543, A323818, A367862, A367863, A367867, A367916, A367917, A368951.

a := n -> `if`(n=0,1,((n-1)*exp(n)*GAMMA(n-1,n)+n^(n-2)*(3-n))/2):
seq(simplify(a(n)),n=0..16); # Peter Luschny, Jan 18 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[ Log[1/(1-t)]/2+t/2-3t^2/4+1,{x,0,nn}],x]  (* Geoffrey Critzer, Mar 23 2013 *)

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + t/2 - 3*t^2/4 + 1))} \\ Andrew Howroyd, Nov 07 2019

a(0) = 1, for n >=1, a(n) = A000272(n) + A057500(n) = n^{n-2} + (n-1)(n-2)/2Sum_{r=1..n-2}( (n-3)!/(n-2-r)! )n^(n-2-r)
E.g.f.: log(1/(1-T(x)))/2 + T(x)/2 - 3*T(x)^2/4 + 1, where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 23 2013
a(n) = ((n-1)*e^n*GAMMA(n-1,n)+n^(n-2)*(3-n))/2 for n>=1. - Peter Luschny, Jan 18 2016

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 2, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 1, 10, 0, 0, 0, 1, 0, 0, 0, 10, 0, 0, 1, 1, 0, 0, 0, 7, 4, 0, 0, 2, 0, 0

Offset: 1

Gus Wiseman, Dec 27 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is counted under a(36).

			The a(1) = 0 through a(24) = 3 factorizations:
 ... 2*2 ... 2*4   3*3 .. 2*2*3 ... 2*8     . 2*3*3 . 2*2*5 ... 2*2*6
             2*2*2                  4*4                         2*3*4
                                    2*2*4                       2*2*2*3
                                    2*2*2*2

For unlabeled graphs: A140637, complement A134964.
For labeled graphs: A367867, A367868, A140638, complement A133686.
For set-systems: A367903, ranks A367907, complement A367902, ranks A367906.
For non-isomorphic set-systems: A368094, A368409, complement A368095.
For non-isomorphic multiset partitions: A368097, A355529, A368411.
Complement for non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368414.
For non-isomorphic set multipartitions: A368421, complement A368422.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]=={}&]],{n,100}]

a(n) + A368414(n) = A001055(n).

A368097 Number of non-isomorphic multiset partitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 3, 12, 37, 133, 433, 1516, 5209, 18555

Offset: 0

Gus Wiseman, Dec 25 2023

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 multiset partitions:
  {{1},{1}}  {{1},{1,1}}    {{1},{1,1,1}}
             {{1},{1},{1}}  {{1,1},{1,1}}
             {{1},{2},{2}}  {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{1},{2,3}}
                            {{1},{2},{1,2}}
                            {{1},{2},{2,2}}
                            {{2},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}
                            {{1},{2},{2},{2}}
                            {{1},{2},{3},{3}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs appears to be A140637, complement A134964.
These multiset partitions have ranks A355529.
The case of labeled graphs is A367867, complement A133686.
Set-systems not of this type are A367902, ranks A367906.
Set-systems of this type are A367903, ranks A367907.
For set-systems we have A368094, complement A368095.
The complement is A368098, ranks A368100, connected case A368412.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368411.
Factorizations of this type are counted by A368413, complement A368414.
For set multipartitions we have A368421, complement A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A316983, A317533, A319616, A321405, A367905, A368409, A368410.

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]}];
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], Select[Tuples[#],UnsameQ@@#&]=={}&]]], {n,0,6}]

A140637 Number of unlabeled graphs of positive excess with n nodes.

Original entry on oeis.org

0, 0, 0, 2, 15, 110, 936, 12073, 273972, 12003332, 1018992968, 165091159269, 50502031331411, 29054155657134165, 31426485969804026075, 64001015704527557101231, 245935864153532932681481794, 1787577725145611700547871854870, 24637809253125004524383007473440146

Offset: 1

Washington Bomfim, May 21 2008

nonn

We can find in "The Birth of the Giant Component" p. 53, see the link, the following: "The excess of a graph or multigraph is the number of edges plus the number of acyclic components, minus the number of vertices."
If G has just one complex component with 4 nodes, the "non-complex part" of G can be,
a) One forest of order 4. There are 6 forests, so 2*6=12 graphs.
b) One triangle and one isolated vertex, or 2*1=2 graphs.
c) One unicyclic graph of order 4, so 2*2=4 graphs.
Also the number of unchoosable unlabeled graphs with up to n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The labeled version is A367867, covering A367868, connected A140638. - Gus Wiseman, Feb 13 2024

			Below we show that a(8) = 12073. Note that A140636(n) is the number of connected graphs of positive excess with n nodes.
Let G be a disconnected graph of positive excess with 8 nodes. In this case, G has one or two complex components. We have 3 graphs of order 8 with two complex components. One of those graphs is depicted in the figure below:
  O---O...O---O
  |\..|...|\./|
  |.\.|...|.X.|
  |..\|...|/.\|
  O---O...O---O
If G has one complex component with 5 nodes, the non-complex part of G can be,
a) One forest of order 3. There are 3 forests, so A140636(5) * 3 = 39 graphs.
b) One triangle, so A140636(5) = 13 graphs.
If G has one complex component with 6 nodes, the non-complex part of G is a forest of order 2. There are 2 forests. We have A140636(6) * 2, or 186 graphs.
If G has one complex component with 7 nodes, the non-complex part of G is one isolated vertex. We have A140636(7), or 809 graphs.
Finally if G is connected, we have A140636(8), or 11005 graphs.
The total is 3 + 12 + 2 + 4 + 39 + 13 + 186 + 809 + 11005 = 12073.

Svante Janson, Donald E. Knuth, Tomasz Łuczak and Boris Pittel, The Birth of the Giant Component.

Cf. A000088, A005195, A001429.
The labeled complement is A133686, covering A367869, connected A129271.
The complement is A134964, connected A005703.
The connected covering case is A140636.
The labeled version is A367867, covering A367868, connected A140638.
Set-systems not of this type are A367902, ranks A367906.
Set-systems of this type are A367903, ranks A367907.
For set-systems we have A368094, complement A368095.
For multiset partitions we have A368097, complement A368098.
Factorizations of this type are A368413, complement A368414.
Cf. A001349, A002494, A006125, A055621, A367769, A367770, A367901.

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}]],Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

a(n) = A000088(n) - A134964(n).

A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is not counted under a(36).

			The a(n) factorizations for selected n:
  1    6      12     24      30       60        72      120
       2*3    2*6    2*12    2*15     2*30      2*36    2*60
              3*4    3*8     3*10     3*20      3*24    3*40
                     4*6     5*6      4*15      4*18    4*30
                             2*3*5    5*12      6*12    5*24
                                      6*10      8*9     6*20
                                      2*3*10            8*15
                                      2*5*6             10*12
                                      3*4*5             2*3*20
                                                        2*5*12
                                                        2*6*10
                                                        3*4*10
                                                        3*5*8
                                                        4*5*6

For labeled graphs: A133686, complement A367867, A367868, A140638.
For unlabeled graphs: A134964, complement A140637.
For set-systems: A367902, ranks A367906, complement A367903, ranks A367907.
For non-isomorphic set-systems: A368095, complement A368094, A368409.
Complementary non-isomorphic multiset partitions: A368097, A355529, A368411.
For non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368413.
For non-isomorphic set multipartitions: A368422, complement A368421.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412, A368413.

facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]],{n,100}]

a(n) = A001055(n) - A368413(n).

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791

Offset: 0

Washington Bomfim, Feb 22 2008

Also the number of labeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. The version without the choice condition is A116508, covering A367863. - Gus Wiseman, Jan 25 2024

			a(6) = 3670 because A057500(6) = 3660 and two triangles can be labeled in 10 ways.
From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(4) = 15 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}
                        {12,13,14,24}
                        {12,13,14,34}
                        {12,13,23,24}
                        {12,13,23,34}
                        {12,13,24,34}
                        {12,14,23,24}
                        {12,14,23,34}
                        {12,14,24,34}
                        {12,23,24,34}
                        {13,14,23,24}
                        {13,14,23,34}
                        {13,14,24,34}
                        {13,23,24,34}
                        {14,23,24,34}
(End)

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 151..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

The connected case is A057500.
Row sums of A106239.
The unlabeled version is A137917.
Diagonal of A144228.
The version with loops appears to be A333331, unlabeled A368984.
Column k = 0 of A368924.
The complement is counted by A369143, unlabeled A369201, covering A369144.
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 simple graphs, covering A367869.
A140637 counts unlabeled non-choosable graphs, covering A369202.
A367867 counts non-choosable graphs, covering A367868.
Cf. A001434, A006649, A053530, A116508, A129271, A134964, A140638, A367862, A367863, A369191.

cy:= proc(n) option remember;
       binomial(n-1, 2)*add((n-3)!/(n-2-t)!*n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember; `if`(k=0, 1, `if`(k<0 or n T(n,n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2008

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

A057500(p) = (p-1)! * p^p /2 * sum(k = 3,p, 1/(p^k*(p-k)!)); /* Vladeta Jovovic, A057500. */
F(n,N) = { my(s = 0, K, D, Mc); forpart(P = n, D = Set(P); K = vector(#D);
for(i=1, #D, K[i] = #select(x->x == D[i], Vec(P)));
Mc = n!/prod(i=1,#D, K[i]!);
s += Mc * prod(i = 1, #D, A057500(D[i])^K[i] / ( D[i]!^K[i])) , [3, n], [N, N]); s };
a(n)= {my(N); sum(N = 1, n, F(n,N))};

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-log(1+w)/2 + w/2 - w^2/4)))} \\ Andrew Howroyd, May 18 2021

a(n) = Sum_{N = 1..n} ((n!/N!) * Sum_{n_1 + n_2 + ... + n_N = n} Product_{i = 1..N} ( A057500(n_i) / n_i! ) ). [V. F. Kolchin p. 31, (1.4.2)] replacing numerator terms n_i^(n_i-2) by A057500(n_i).
a(n) = A144228(n,n). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x) = (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012
a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013
E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A057500. - Andrew Howroyd, May 18 2021

a(0)=1 prepended by Andrew Howroyd, May 18 2021

A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 12, 36, 97, 291

Offset: 0

Gus Wiseman, Dec 23 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(5) = 1 through a(7) = 12 set-systems:
  {{1},{2},{3},{2,3}}  {{1},{2},{1,3},{2,3}}    {{1},{2},{1,2},{3,4,5}}
                       {{1},{2},{3},{1,2,3}}    {{1},{3},{2,3},{1,2,3}}
                       {{2},{3},{1,3},{2,3}}    {{1},{4},{1,4},{2,3,4}}
                       {{3},{4},{1,2},{3,4}}    {{2},{3},{2,3},{1,2,3}}
                       {{1},{2},{3},{4},{3,4}}  {{3},{1,2},{1,3},{2,3}}
                                                {{1},{2},{3},{1,3},{2,3}}
                                                {{1},{2},{3},{2,4},{3,4}}
                                                {{1},{2},{3},{4},{2,3,4}}
                                                {{1},{3},{4},{2,4},{3,4}}
                                                {{1},{4},{5},{2,3},{4,5}}
                                                {{2},{3},{4},{1,2},{3,4}}
                                                {{1},{2},{3},{4},{5},{4,5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
The case of labeled graphs is A367867, complement A133686.
The labeled version is A367903, ranks A367907.
The complement is counted by A368095, connected A368410.
Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368409.
Factorizations of this type are counted by A368413, complement A368414.
Allowing repeated edges gives A368421, complement A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A316983, A317533, A319616, A321155, A321405, A326031, A330223, A330227, A367905.

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]}];
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], UnsameQ@@#&&And@@UnsameQ@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,8}]

A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 86, 208, 508, 1304

Offset: 0

Gus Wiseman, Dec 24 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 17 set-systems:
  {1}  {12}    {123}      {1234}        {12345}
       {1}{2}  {1}{23}    {1}{234}      {1}{2345}
               {2}{12}    {12}{34}      {12}{345}
               {1}{2}{3}  {13}{23}      {14}{234}
                          {3}{123}      {23}{123}
                          {1}{2}{34}    {4}{1234}
                          {1}{3}{23}    {1}{2}{345}
                          {1}{2}{3}{4}  {1}{23}{45}
                                        {1}{24}{34}
                                        {1}{4}{234}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {4}{12}{34}
                                        {1}{2}{3}{45}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

For labeled graphs we have A133686, complement A367867.
For unlabeled graphs we have A134964, complement A140637.
For set-systems we have A367902, complement A367903.
These set-systems have BII-numbers A367906, complement A367907.
The complement is A368094, connected A368409.
Repeats allowed: A368098, ranks A368100, complement A368097, ranks A355529.
Minimal multiset partitions not of this type are counted by A368187.
The connected case is A368410.
Factorizations of this type are counted by A368414, complement A368413.
Allowing repeated edges gives A368422, complement A368421.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A001055, A007717, A302545, A306005, A316983, A319560, A321194, A321405, A330223, A367769, A367901.

Table[Length[Select[bmp[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]], {n,0,10}]

A005703 Number of n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 19, 44, 112, 287, 763, 2041, 5577, 15300, 42419, 118122, 330785, 929469, 2621272, 7411706, 21010378, 59682057, 169859257, 484234165, 1382567947, 3952860475, 11315775161, 32430737380, 93044797486, 267211342954, 768096496093, 2209772802169

Offset: 0

N. J. A. Sloane, R. K. Guy

a(n) is the number of pseudotrees on n nodes. - Eric W. Weisstein, Jun 11 2012

Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024

			From _Gus Wiseman_, Feb 20 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 8 graphs:
  {}  .  {12}  {12,13}     {12,13,14}     {12,13,14,15}
               {12,13,23}  {12,13,24}     {12,13,14,25}
                           {12,13,14,23}  {12,13,24,35}
                           {12,13,24,34}  {12,13,14,15,23}
                                          {12,13,14,23,25}
                                          {12,13,14,23,45}
                                          {12,13,14,25,35}
                                          {12,13,24,35,45}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Washington Bomfim, Table of n, a(n) for n = 0..500
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Pseudotree
Wikipedia, Pseudoforest

Cf. A000055, A000081, A001429 (labeled A057500), A134964 (number of pseudoforests, labeled A133686).
The labeled version is A129271.
The connected complement is A140636, labeled A140638.
Non-connected: A368834 (labeled A367869) or A370316 (labeled A369191).
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A062734 counts connected graphs by number of edges.
Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, A368951, A369197, A370317, A370318.

Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}];
a[0] = 0;
b = Drop[Flatten[
    sol = SolveAlways[
      0 == Series[
        t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
      x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
    CoefficientList[
     Series[CycleIndex[DihedralGroup[n], s] /.
       Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
     nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; CoefficientList[
Series[r[x] - (r[x]^2 - r[x^2])/2 + d[x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 17 2014 *)

\\ TreeGf gives gf of A000081.
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)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + g(1) + (g(2) - g(1)^2)/2 + sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2)}; \\ Andrew Howroyd and Washington Bomfim, May 15 2021

a(n) = A000055(n) + A001429(n).

More terms from Vladeta Jovovic, Apr 19 2000 and from Michael Somos, Apr 26 2000

a(27) corrected and a(28) and a(29) computed by Washington Bomfim, May 14 2008

cp's OEIS Frontend

A133686 Number of labeled n-node graphs with at most one cycle in each connected component.

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A129271 Number of labeled n-node connected graphs with at most one cycle.

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A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.

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A368097 Number of non-isomorphic multiset partitions of weight n contradicting a strict version of the axiom of choice.

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A140637 Number of unlabeled graphs of positive excess with n nodes.

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A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

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A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

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