A003465 -id:A003465 - cp's OEIS frontend

A326878 Number of topologies whose points are a subset of {1..n}.

1, 2, 7, 45, 500, 9053, 257151, 11161244, 725343385, 69407094565, 9639771895398, 1919182252611715, 541764452276876719, 214777343584048313318, 118575323291814379721651, 90492591258634595795504697, 94844885130660856889237907260, 135738086271526574073701454370969, 263921383510041055422284977248713291

Offset: 0

Gus Wiseman, Jul 30 2019

nonn

			The a(0) = 1 through a(2) = 7 topologies:
  {{}}  {{}}      {{}}
        {{},{1}}  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Wikipedia Topological space

Binomial transform of A000798 (the covering case).

Cf. A001035, A001930, A003465, A014466, A102896, A102897, A306445, A326866, A326876.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}]
(* Second program: *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
a[n_] := Sum[Binomial[n, k]*A000798[[k+1]], {k, 0, n}];
a /@ Range[0, Length[A000798]-1] (* Jean-François Alcover, Dec 30 2019 *)

From Geoffrey Critzer, Jul 12 2022: (Start)
E.g.f.: exp(x)*A(exp(x)-1) where A(x) is the e.g.f. for A001035.
a(n) = Sum_{k=0..n} binomial(n,k)*A000798(k). (End)

A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 4, 4, 4, 4, 7, 7, 7, 7, 3, 3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8

Offset: 0

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. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

The run-lengths are all 4 or 8.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with multiset choices {1,1,2}, {1,1,3}, {1,2,2}, {1,2,3}, {1,3,3}, {2,2,3}, {2,3,3}, so a(52) = 7.

Positions of ones are A253317.
The version for multisets and divisors is A355733, for sequences A355731.
The version for multisets is A355744, for sequences A355741.
For a sequence of distinct choices we have A367905, firsts A367910.
Positions of first appearances are A367913, sorted A367915.
Choosing a sequence instead of multiset gives A368109, firsts A368111.
Choosing a set instead of multiset gives A368183, firsts A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355735, A355739, A355740, A355745, A367771, A367906.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]], {n,0,100}]

A046165 Number of minimal covers of n objects.

Original entry on oeis.org

1, 1, 2, 8, 49, 462, 6424, 129425, 3731508, 152424420, 8780782707, 710389021036, 80610570275140, 12815915627480695, 2855758994821922882, 892194474524889501292, 391202163933291014701953, 240943718535427829240708786, 208683398342300491409959279244

Offset: 0

Eric W. Weisstein

nonn

No edge of a minimal cover can be a subset of any other, so minimal covers are antichains, but the converse is not true. - Gus Wiseman, Jul 03 2019

a(n) is the number of undirected graphs on n nodes for which the intersection number and independence number are equal. See Proposition 2.3.7 and Theorem 2.3.3 of the Deligeorgaki et al. paper below. - Alex Markham, Oct 13 2022

			From _Gus Wiseman_, Jul 02 2019: (Start)
The a(1) = 1 through a(3) = 8 minimal covers:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1,2},{1,3}}
                    {{1,2},{2,3}}
                    {{1},{2},{3}}
                    {{1,3},{2,3}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..113
Damian Bursztyn, François Goasdoué, and Ioana Manolescu, Optimizing Reformulation-based Query Answering in RDF, [Research Report] RR-8646, INRIA Saclay. 2014.
D. Deligeorgaki, A. Markham, P. Misra, and L. Solus, Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks, arXiv:2210.00822 [stat.ME], 2022.
Giovanni Resta, Illustration of a(4)=49.
Eric Weisstein's World of Mathematics, Minimal Cover

Cf. A035348, A000371, A003465.
Antichain covers are A006126.
Minimal covering simple graphs are A053530.
Maximal antichains are A326358.
Row sums of A035347 or of A282575.
Cf. A000372, A003182, A006602, A261005, A305844, A307249, A326359.

a:= n-> add(add((-1)^i* binomial(k,i) *(2^k-1-i)^n, i=0..k)/k!, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 19 2008

Table[Sum[Sum[Binomial[n,i]StirlingS2[i,k](2^k-k-1)^(n-i),{i,k,n}],{k,2,n}]+1,{n,1,20}] (* Geoffrey Critzer, Jun 27 2013 *)

E.g.f.: Sum_{n>=0} (exp(x)-1)^n*exp(x*(2^n-n-1))/n!. - Vladeta Jovovic, May 08 2004
a(n) = Sum_{k=1..n} Sum_{i=k..n} C(n,i)*Stirling2(i,k)*(2^k - k - 1)^(n - i). - Geoffrey Critzer, Jun 27 2013
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

a(0)=1 prepended by Alois P. Heinz, Feb 18 2017

A326876 BII-numbers of finite topologies without their empty set.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 16, 17, 24, 25, 32, 34, 40, 42, 64, 65, 66, 68, 69, 70, 71, 72, 76, 80, 81, 82, 85, 87, 88, 89, 93, 96, 97, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 256, 257, 384, 385, 512, 514, 640, 642, 1024, 1025, 1026, 1028, 1029, 1030

Offset: 1

Gus Wiseman, Jul 29 2019

nonn

A finite topology is a finite set of finite sets closed under union and intersection and containing {} and the vertex set.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. 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 enumeration of finite topologies by number of points is given by A000798.

			The sequence of all finite topologies without their empty set together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}

Wikipedia Topological space

Cf. A000798, A001930, A003465, A048793, A102894, A102896, A326031, A326872, A326875, A326878.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]

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

A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 25, 67, 134, 309, 709, 1579, 3420, 7240, 15077, 30997, 61994, 125364, 253712, 512411, 1032453, 2075737, 4166469, 8352851, 16731873, 33497422, 67038086, 134130344, 268328977, 536741608, 1073586022, 2147296425, 4294592850, 8589346462, 17179033384

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 0 through a(5) = 8 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,2,3,4}  {1,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370582, differences A370586.
For prime indices we have A370583, differences A370587.
First differences are A370589.
The complement is counted by A370636, differences A370639.
The case without ones is A370643.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The minimal case is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

a(2^n - 1) = A367903(n).

Partial sums of A370589.

a(21)-a(34) from Alois P. Heinz, Mar 09 2024

A094574 Number of (<=2)-covers of an n-set.

Original entry on oeis.org

1, 1, 5, 40, 457, 6995, 136771, 3299218, 95668354, 3268445951, 129468914524, 5868774803537, 301122189141524, 17327463910351045, 1109375488487304027, 78484513540137938209, 6098627708074641312182, 517736625823888411991202, 47791900951140948275632148

Offset: 0

Goran Kilibarda, Vladeta Jovovic, May 12 2004

nonn

Also the number of strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. For example, the a(2) = 5 strict multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (11)(22), (1)(2)(12). - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Sep 02 2019: (Start)
These are set-systems covering {1..n} with vertex-degrees <= 2. For example, the a(3) = 40 covers are:
  {123}  {1}{23}    {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}    {1}{2}{13}    {1}{2}{3}{13}
         {3}{12}    {1}{2}{23}    {1}{2}{3}{23}
         {1}{123}   {1}{3}{12}    {1}{2}{13}{23}
         {12}{13}   {1}{3}{23}    {1}{2}{3}{123}
         {12}{23}   {2}{3}{12}    {1}{3}{12}{23}
         {13}{23}   {2}{3}{13}    {2}{3}{12}{13}
         {2}{123}   {1}{12}{23}
         {3}{123}   {1}{13}{23}
         {12}{123}  {1}{2}{123}
         {13}{123}  {1}{3}{123}
         {23}{123}  {2}{12}{13}
                    {2}{13}{23}
                    {2}{3}{123}
                    {3}{12}{13}
                    {3}{12}{23}
                    {12}{13}{23}
                    {1}{23}{123}
                    {2}{13}{123}
                    {3}{12}{123}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A219585. - Alois P. Heinz, Nov 23 2012
Dominated by A003465.
Graphs with vertex-degrees <= 2 are A136281.
Cf. A002718, A007716, A020554, A020555, A050535, A094574, A136284, A316974, A327104, A327106, A327229.
Main diagonal of A346517.

facs[n_]:=facs[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[Array[Prime,n,1,Times]^2],UnsameQ@@#&]],{n,0,6}] (* Gus Wiseman, Jul 18 2018 *)
m = 20;
a094577[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
CoefficientList[egf + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, May 13 2019 *)

Row sums of A094573.

E.g.f: exp(-1-1/2*(exp(x)-1))*Sum(exp(x*binomial(n+1, 2))/n!, n=0..infinity) or exp((1-exp(x))/2)*Sum(A094577 (n)*(x/2)^n/n!, n=0..infinity).

A186021 a(n) = Bell(n)*(2 - 0^n).

Original entry on oeis.org

1, 2, 4, 10, 30, 104, 406, 1754, 8280, 42294, 231950, 1357140, 8427194, 55288874, 381798644, 2765917090, 20960284294, 165729739608, 1364153612318, 11665484410114, 103448316470744, 949739632313502, 9013431476894646, 88304011710168692, 891917738589610578, 9277180664459998706

Offset: 0

Paul Barry, Feb 10 2011

a(n) is the number of collections of subsets of {1,2,...,n-1} that are pairwise disjoint. a(n+1) = 2*Sum_{j=0..n} C(n,j)*Bell(j). For example a(3)=10 because we have: {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{},{1}}, {{},{2}}, {{},{1,2}}, {{1},{2}}, {{},{1},{2}}. - Geoffrey Critzer, Aug 28 2014

a(n) is the number of collections of subsets of [n] that are pairwise disjoint and cover [n], with [0] = {}. For disjoint collections of nonempty subsets see A000110. For arbitrary collections of subsets see A000371. For arbitrary collections of nonempty subsets see A003465. - Manfred Boergens, May 02 2024 and Apr 09 2025

			a(4) = A060719(3) + 1 = 29 + 1 = 30.

G. C. Greubel, Table of n, a(n) for n = 0..575

Row sums of A186020 and A256894.
Main diagonal of A271466 (shifted) and A381682.
Cf. A000110, A000371, A003465.

[Bell(n)*(2-0^n): n in [0..50]]; // Vincenzo Librandi, Apr 06 2011

A186021List := proc(m) local A, P, n; A := [1,2]; P := [2];
for n from 1 to m - 2 do P := ListTools:-PartialSums([P[-1], op(P)]);
A := [op(A), P[-1]] od; A end: A186021List(26); # Peter Luschny, Mar 24 2022

Prepend[Table[2 Sum[Binomial[n, j] BellB[j], {j, 0, n}], {n, 0, 25}], 1] (* Geoffrey Critzer, Aug 28 2014 *)
With[{nmax = 50}, CoefficientList[Series[2*Exp[Exp[x] - 1] - 1, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Jul 24 2017 *)

x='x+O('x^50); Vec(serlaplace(2*exp(exp(x) - 1) -1)) \\ G. C. Greubel, Jul 24 2017

from itertools import accumulate
def A186021_list(size):
    if size < 1: return []
    L, accu = [1], [2]
    for _ in range(size-1):
        accu = list(accumulate([accu[-1]] + accu))
        L.append(accu[0])
    return L
print(A186021_list(26)) # Peter Luschny, Apr 25 2016

E.g.f.: 2*exp(exp(x)-1)-1. - Paul Barry, Apr 06 2011
a(n) = A000110(n)*A040000(n).
a(n+1) = 1 + Sum_{k=0..n} C(n,k)*a(k). - Franklin T. Adams-Watters, Oct 02 2011
From Sergei N. Gladkovskii, Nov 11 2012 to Mar 29 2013: (Start)
Continued fractions:
G.f.: A(x)= 1 + 2*x/(G(0)-x) where G(k)= 1 - x*(k+1)/(1 - x/G(k+1)).
G.f.: G(0)-1 where G(k) = 1-(x*k+1)/(x*k - 1 - x*(x*k - 1)/(x + (x*k + 1)/G(k+1))).
G.f.: (G(0)-2)/x - 1 where G(k) = 1 + 1/(1 - x/(x + (1 - x*k)/G(k+1))).
G.f.: (S-2)/x - 1 where S = 2*Sum_{k>=0} x^k/Product_{n=0..k-1}(1 - n*x).
G.f.: 1/(1-x) - x/(G(0)-x^2+x) where G(k) =x^2 + x - 1 + k*(2*x-x^2) - x^2*k^2 + x*(x*k - 1)*(x*k + 2*x - 1)^2/G(k+1).
E.g.f.: E(0) - 1 where E(k) = 1 + 1/(1 - 1/(1 + (k+1)/x*Bell(k)/Bell(k+1)/E(k+1))). (End)
a(n) = A060719(n-1) + 1, and the inverse binomial transform of A060719. - Gary W. Adamson, May 20 2013
G.f. A(x) satisfies: A(x) = 1 + (x/(1 - x)) * (1 + A(x/(1 - x))). - Ilya Gutkovskiy, Jun 30 2020

A006058 Number of connected labeled T_4-topologies with n points.

Original entry on oeis.org

1, 1, 3, 16, 145, 2111, 47624, 1626003, 82564031, 6146805142, 662718022355, 102336213875523, 22408881211102698, 6895949927379360277, 2958271314760111914191, 1756322140048351303019576

Offset: 0

N. J. A. Sloane

From Gus Wiseman, Aug 05 2019: (Start)
For n > 0, also the number of topologies covering {1..n} whose nonempty open sets have nonempty intersection. Also the number of topologies covering {1..n} whose nonempty open sets are pairwise intersecting. For example, the a(0) = 1 through a(3) = 16 topologies (empty sets not shown) are:
  {}  {{1}}  {{1,2}}      {{1,2,3}}
             {{1},{1,2}}  {{1},{1,2,3}}
             {{2},{1,2}}  {{2},{1,2,3}}
                          {{3},{1,2,3}}
                          {{1,2},{1,2,3}}
                          {{1,3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Herman Jamke, Table of n, a(n) for n = 0..19
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

Cf. A001930, A003465, A108798, A306445, A326878, A326906.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4}] (* Gus Wiseman, Aug 05 2019 *)
A000798 = Append[Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]], 0];
a[n_] := If[n == 0, 1, Sum[ Binomial[n, k] A000798[[k+1]], {k, 0, n-1}]];
a /@ Range[0, Length[A000798]-1] (* Jean-François Alcover, Jan 01 2020 *)

From Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008: (Start)
a(n) = Sum_{k=0..n-1} binomial(n, k)*A000798(k) if n>=1.
E.g.f.: Z4(x) = A(x)*(exp(x)-1) + 1 where A(x) denotes the e.g.f. for A000798. (End)
a(n) = A326909(n) - A000798(n). - Gus Wiseman, Aug 05 2019

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

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

cp's OEIS Frontend

A326878 Number of topologies whose points are a subset of {1..n}.

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A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

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A046165 Number of minimal covers of n objects.

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Maple

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A326876 BII-numbers of finite topologies without their empty set.

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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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A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

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A094574 Number of (<=2)-covers of an n-set.

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A186021 a(n) = Bell(n)*(2 - 0^n).

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