A367867 -id:A367867 - 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

A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 67, 30997, 2147296425, 9223372036784737528, 170141183460469231731687303625772608225, 57896044618658097711785492504343953926634992332820282019728791606173188627779

Offset: 0

Gus Wiseman, Dec 05 2023

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 a(2) = 1 set-system is {{1},{2},{1,2}}.
The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
  {{1},{2},{1,2}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{1,3},{1,2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Wikipedia, Axiom of choice.

Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version for simple graphs is A367867, covering A367868.
The version allowing empty edges is A367901.
The complement is A367902, without singletons A367770, ranks A367906.
For a unique choice (instead of none) we have A367904, ranks A367908.
These set-systems have ranks A367907.
An unlabeled version is A368094, for multiset partitions A368097.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]

a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A239312 Number of condensed integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044

Offset: 0

Clark Kimberling, Mar 15 2014

nonn

Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.

Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024

			a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
From _Gus Wiseman_, Mar 12 2024: (Start)
The a(1) = 1 through a(9) = 10 condensed partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                   (3,1)  (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,2,2)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,2,2)  (4,4,1)
                                                   (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..100 (first 84 terms from Manfred Scheucher)
Manfred Scheucher, Python Script

The strict case is A000009.
These partitions have ranks A368110, complement A355740.
The complement is counted by A370320.
The version for prime factors (not all divisors) is A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370805, complement A370804.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions.
A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.

b:= proc(n,i) option remember; `if`(n=0, {[]},
      `if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
       sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> nops(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 01 2019

u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0,   30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]], {n,0,30}] (* Gus Wiseman, Mar 12 2024 *)

Typo in definition corrected by Manfred Scheucher, May 29 2015

Name edited by Gus Wiseman, Mar 13 2024

A116508 a(n) = C( C(n,2), n).

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 5005, 116280, 3108105, 94143280, 3190187286, 119653565850, 4922879481520, 220495674290430, 10682005290753420, 556608279578340080, 31044058215401404845, 1845382436487682488000, 116475817125419611477660, 7779819801401934344268210

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Mar 21 2006

a(n) is the number of simple labeled graphs with n nodes and n edges. - Geoffrey Critzer, Nov 02 2014

These graphs are not necessarily covering, but the covering case is A367863, unlabeled A006649, and the unlabeled version is A001434. - Gus Wiseman, Dec 22 2023

			a(5) = C(C(5,2),5) = C(10,5) = 252.

Alois P. Heinz, Table of n, a(n) for n = 0..370

Main diagonal of A084546.
The unlabeled version is A001434, covering case A006649.
The connected case is A057500, unlabeled A001429.
For set-systems we have A136556, covering case A054780.
The covering case is A367863.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A133686 counts graphs satisfying a strict AOC, connected A129271.
A367867 counts graphs contradicting a strict AOC, connected A140638.
Cf. A001187, A003465, A143543, A305000, A367916, A367917.

[0] cat [(Binomial(Binomial(n+2, n), n+2)): n in [0..20]]; // Vincenzo Librandi, Nov 03 2014

a:= n-> binomial(binomial(n, 2), n):
seq(a(n), n=0..20);

nn = 18; f[x_, y_] :=
Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 1, nn}]; Table[
n! Coefficient[Series[f[x, y], {x, 0, nn}], x^n y^n], {n, 1, nn}] (* Geoffrey Critzer, Nov 02 2014 *)
Table[Length[Subsets[Subsets[Range[n],{2}],{n}]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)
Table[SeriesCoefficient[(1 + x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

from math import comb
def A116508(n): return comb(n*(n-1)>>1,n) # Chai Wah Wu, Jul 02 2024

[(binomial(binomial(n+2,n),n+2)) for n in range(-1, 17)] # Zerinvary Lajos, Nov 30 2009

a(n) ~ exp(n - 2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2020

a(n) = [x^n] (1+x)^(n*(n-1)/2). - Vaclav Kotesovec, Aug 06 2025

a(0)=1 prepended by Alois P. Heinz, Feb 02 2024

A367863 Number of n-vertex labeled simple graphs with n edges and no isolated vertices.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3760, 73755, 1657845, 42143500, 1197163134, 37613828070, 1295741321875, 48577055308320, 1969293264235635, 85852853154670693, 4005625283891276535, 199166987259400191480, 10513996906985414443720, 587316057411626070658200, 34612299496604684775762261

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

			Non-isomorphic representatives of the a(4) = 15 graphs:
  {{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..200

The connected case is A057500, unlabeled A001429.
The unlabeled version is A006649.
The non-covering version is A116508.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A003465, A006126, A305000, A316983, A319559, A323817, A326754, A367769, A367901, A367902, A367903.

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

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform is A367862.

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n). - Andrew Howroyd, Dec 29 2023

Terms a(8) and beyond from Andrew Howroyd, Dec 29 2023

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

A367869 Number of labeled simple graphs covering n vertices and satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5596, 97149, 1959938, 44956945, 1154208544, 32772977715, 1019467710328, 34473686833527, 1259038828370402, 49388615245426933, 2070991708598960524, 92445181295983865757, 4376733266230674345874, 219058079619119072854095, 11556990682657196214302036

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

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.

Number of labeled n-node graphs with at most one cycle in each component and no isolated vertices. - Andrew Howroyd, Dec 30 2023

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

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

The connected case is A129271.
The non-covering case is A133686, complement A367867.
The complement is A367868, connected A140638 (unlabeled A140636).
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A355740, A367770, A367863, A367902.

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

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(sqrt(1/(1-t))*exp(t/2 - 3*t^2/4 - x)))} \\ Andrew Howroyd, Dec 30 2023

E.g.f.: exp(B(x) - x - 1) where B(x) is the e.g.f. of A129271. - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367868 Number of labeled simple graphs covering n vertices and contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 381, 21853, 1790135, 250562543, 66331467215, 34507857686001, 35645472109753873, 73356936892660012513, 301275024409580265134121, 2471655539736293803311467943, 40527712706903494712385171632959, 1328579255614092966328511889576785109

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

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 a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{3,4}}
  {{1,2},{1,3},{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

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

The connected case is A140638, unlabeled A140636.
The non-covering case is A367867.
The complement is A367869, connected A129271, non-covering A133686.
The version for set-systems is A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A367769, A367770, A367863, A367901, A367902, A367904.

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

a(n) = A006129(n) - A367869(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

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}]

cp's OEIS Frontend

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

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A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.

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A239312 Number of condensed integer partitions of n.

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A116508 a(n) = C( C(n,2), n).

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A367863 Number of n-vertex labeled simple graphs with n edges and no isolated vertices.

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

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