A367901 -id:A367901 - cp's OEIS frontend

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

1, 1, 1, 2, 20, 308, 5338, 105298, 2366704, 60065072, 1702900574, 53400243419, 1836274300504, 68730359299960, 2782263907231153, 121137565273808792, 5645321914669112342, 280401845830658755142, 14788386825536445299398, 825378055206721558026931, 48604149005046792753887416

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

Unlike the connected case (A057500), these graphs may have more than one cycle; for example, the graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}} has multiple cycles.

			Non-isomorphic representatives of the a(4) = 20 graphs:
  {}
  {{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..200

The connected case is A057500, unlabeled A001429.
Counting all vertices (not just covered) gives A116508.
The covering case is A367863, unlabeled A006649.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A003465 counts covering set-systems, unlabeled A055621, ranks A326754.
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. A006126, A316983, A321405, A326754, A326870, A326877, A367770, A367901, A367902, A367903.

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

\\ Here b(n) is A367863(n)
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A367863.

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

A367904 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a sequence of different vertices of each edge.

Original entry on oeis.org

1, 2, 6, 38, 666, 32282, 3965886, 1165884638, 792920124786, 1220537093266802, 4187268805038970806, 31649452354183112810198, 522319168680465054600480906, 18683388426164284818805590810122, 1439689660962836496648920949576152046, 237746858936806624825195458794266076911118

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The set-system Y = {{1},{1,2},{2,3}} has choices (1,1,2), (1,1,3), (1,2,2), (1,2,3), of which only (1,2,3) has all different elements, so Y is counted under a(3).
The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Christian Sievers, Table of n, a(n) for n = 0..77

The maximal case (n subsets) is A003024.
The version for at least one choice is A367902.
The version for no choices is A367903, no singletons A367769, ranks A367907.
These set-systems have ranks A367908, nonzero A367906.
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, unlabeled A323819.
Cf. A102896, A133686, A283877, A306445, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#],UnsameQ@@#&]]==1&]],{n,0,3}]

a(n) = A367902(n) - A367772(n). - Christian Sievers, Jul 26 2024

Binomial transform of A003024. - Christian Sievers, Aug 12 2024

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

More terms from Christian Sievers, Aug 12 2024

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}]

A140638 Number of connected graphs on n labeled nodes that contain at least two cycles.

Original entry on oeis.org

0, 0, 0, 7, 381, 21748, 1781154, 249849880, 66257728763, 34495508486976, 35641629989151608, 73354595357480683904, 301272202621204113362497, 2471648811029413368450098688, 40527680937730440155535277704046, 1328578958335783199341353852258282496

Offset: 1

Washington Bomfim, May 21 2008

nonn

These are the connected graphs that are neither trees nor unicyclic.

Also connected non-choosable graphs covering n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The unlabeled version is A140636. The complement is counted by A129271. - Gus Wiseman, Feb 20 2024

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

Andrew Howroyd, Table of n, a(n) for n = 1..50

The unlabeled version is A140636.
Cf. A000272 (trees), A001187 (connected graphs), A057500 (connected unicyclic graphs).
The complement is counted by A129271, unlabeled A005703.
The non-connected complement is A133686, covering A367869.
The non-connected version is A367867, unlabeled A140637.
The non-connected covering version is A367868.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A143543 counts simple labeled graphs by number of connected components.
Cf. A001349, A116508, A355740, A367769, A367863, A367901, A367903, A370317.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,5}] (* Gus Wiseman, Feb 19 2024 *)

seq(n)={my(A=O(x*x^n), t=-lambertw(-x + A)); Vec(serlaplace( log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, A)) - log(1/(1-t))/2 - t/2 + 3*t^2/4), -n)} \\ Andrew Howroyd, Jan 15 2022

a(n) = A001187(n) - A129271(n).

a(n) = A001187(n) - A000272(n) - A057500(n).

Definition clarified by Andrew Howroyd, Jan 15 2022

A367771 Number of ways to choose a different prime index of each prime index of 2n + 1.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 0, 2, 0, 0, 2, 1, 1, 2, 3, 1, 1, 2, 0, 2, 0, 1, 4, 1, 0, 1, 3, 0, 1, 1, 2, 3, 2, 0, 2, 2, 0, 1, 1, 1, 4, 2, 1, 3, 2, 0, 2, 3, 0, 3, 1, 1, 3, 0, 0, 2, 0, 1, 0, 1, 1, 5, 0, 0, 2, 2, 2, 2, 2, 0, 2, 4, 0, 1, 1, 0, 4, 2, 1, 2, 2, 0, 4

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of prime indices of 427 = 2*213 + 1 are {{1,1},{1,2,2}}, with four ways to choose (1,2), so a(213) = 4.
The prime indices of prime indices of 1469 = 2*734 + 1 are {{1,2},{1,2,3}}, with four choices (1,2), (1,3), (2,1), (2,3), so a(734) = 4.

The "extended" version below includes alternating zeros at even positions.
Extended positions of zeros are A355529, binary A367907.
The extended version for binary indices is A367905.
Extended positions of nonzeros are A368100, binary A367906.
Extended positions of ones are A368101, binary A367908.
The extended version without distinctness is A355741, for multisets A355744.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A007716, A355739, A355740, A355745, A367901, A367902, A367903.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[prix/@prix[2n+1]], UnsameQ@@#&]],{n,0,100}]

A368596 Number of n-element sets of singletons or pairs of distinct elements of {1..n}, or loop graphs with n edges, such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 66, 1380, 31460, 800625, 22758918, 718821852, 25057509036, 957657379437, 39878893266795, 1799220308202603, 87502582432459584, 4566246347310609247, 254625879822078742956, 15115640124974801925030, 952050565540607423524658, 63425827673509972464868323

Offset: 0

Gus Wiseman, Jan 04 2024

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(3) = 3 set-systems:
  {{1},{2},{1,2}}
  {{1},{3},{1,3}}
  {{2},{3},{2,3}}

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

The version without the choice condition is A014068, covering A368597.
The complement appears to be A333331.
For covering pairs we have A367868.
Allowing edges of any positive size gives A368600, any length A367903.
The covering case is A368730.
The unlabeled version is A368835.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts covering half-loop-graphs, connected A062740.
A369141 counts non-choosable loop-graphs, covering A369142.
A369146 counts unlabeled non-choosable loop-graphs, covering A369147.
Cf. A000272, A000666, A057500, A129271, A133686, A367769, A367863, A367867, A367869, A367901, A367907, A368097, A369199.

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

Terms a(7) and beyond from Andrew Howroyd, Jan 10 2024

A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 1, 1490, 67027582, 144115188036455750, 1329227995784915872903806998967001298, 226156424291633194186662080095093570025917938800079226639565284090686126876

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.

Includes all set-systems with more edges than covered vertices, but this condition is not sufficient.

			The a(3) = 1 set-system is: {{1,2},{1,3},{2,3},{1,2,3}}.

Wikipedia, Axiom of choice.

Set-systems without singletons are counted by A016031, covering A323816.
The complement is A367770, with singletons allowed A367902 (ranks A367906).
The version for simple graphs is A367867, covering A367868.
The version allowing singletons and empty edges is A367901.
The version allowing singletons is A367903, ranks A367907.
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.
Cf. A007716, A092918, A102896, A283877, A306445, A326031, A355739, A355740, A355741, A367904, A367905.

Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n,0,3}]

a(n) = 2^(2^n-n-1) - A367770(n) = A016031(n+1) - A367770(n). - Christian Sievers, Jul 28 2024

a(6)-a(8) from Christian Sievers, Jul 28 2024

A367770 Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 15, 558, 81282, 39400122, 61313343278, 309674769204452

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.

Excludes all set-systems with more edges than covered vertices, but this condition is not sufficient.

			The a(3) = 15 set-systems:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{2,3}}
  {{1,3},{1,2,3}}
  {{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}
  {{1,2},{2,3},{1,2,3}}
  {{1,3},{2,3},{1,2,3}}

Wikipedia, Axiom of choice.

Set-systems without singletons are counted by A016031, covering A323816.
The version for simple graphs is A133686, covering A367869.
The complement is counted by A367769.
The complement allowing singletons and empty sets is A367901.
Allowing singletons gives A367902, ranks A367906.
The complement allowing singletons is A367903, ranks A367907.
These set-systems have ranks A367906 /\ A326781.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A059201, A083323, A092918, A102896, A283877, A305000, A306445, A355739, A355740, A367904, A367905.

Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,3}]

a(6)-a(8) from Christian Sievers, Jul 28 2024

A368600 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 164, 18625, 5491851, 4649088885, 12219849683346

Offset: 0

Gus Wiseman, Jan 01 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.

			The a(3) = 3 set-systems:
  {{1},{2},{1,2}}
  {{1},{3},{1,3}}
  {{2},{3},{2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367903, ranks A367907, no singletons A367769.
The complement is A368601, any length A367902 (see also A367770, A367906).
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, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367868, A367901, A368094, A368097.

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

from itertools import combinations, product, chain
from scipy.special import comb
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(int(comb(2**n-1,n) - a(n))) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) = A136556(n) - A368601(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

A368601 Number of ways to choose a set of n nonempty subsets of {1..n} such that it is possible to choose a different element from each.

Original entry on oeis.org

1, 1, 3, 32, 1201, 151286, 62453670, 84707326890, 384641855115279

Offset: 0

Gus Wiseman, Jan 01 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.

			The a(2) = 3 set-systems:
  {{1},{2}}
  {{1},{1,2}}
  {{2},{1,2}}
Non-isomorphic representatives of the a(3) = 32 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For a unique choice we have A003024, any length A367904 (ranks A367908).
Sets of n nonempty subsets of {1..n} are counted by A136556.
For any length we have A367902, ranks A367906, no singletons A367770.
The complement is A368600, any length A367903 (see also A367907, A367769).
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, unlabeled A323819.
Cf. A003025, A088957, A133686, A334282, A355529, A355740, A367862, A367867, A367901, A367905, A368094, A368097.

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

from itertools import combinations, product, chain
def v(c):
    for elements in product(*c):
        if len(set(elements)) == len(elements):
            return True
    return False
def a(n):
    if n == 0:
        return 1
    subsets = list(chain.from_iterable(combinations(range(1, n + 1), r) for r in
range(1, n + 1)))
    cs = combinations(subsets, n)
    c = sum(1 for c in cs if v(c))
    return c
[print(a(n)) for n in range(7)] # Robert P. P. McKone, Jan 02 2024

a(n) + A368600(n) = A136556(n).

a(6) from Robert P. P. McKone, Jan 02 2024

a(7)-a(8) from Christian Sievers, Jul 25 2024

cp's OEIS Frontend

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

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A367904 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a sequence of different vertices of each edge.

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

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A140638 Number of connected graphs on n labeled nodes that contain at least two cycles.

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A367771 Number of ways to choose a different prime index of each prime index of 2n + 1.

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A368596 Number of n-element sets of singletons or pairs of distinct elements of {1..n}, or loop graphs with n edges, such that it is not possible to choose a different element from each.

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A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

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A367770 Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.