A001930 -id:A001930 - cp's OEIS frontend

A108800 Number of nonisomorphic systems enumerated by A102895.

1, 2, 6, 28, 330, 28960, 216562364, 5592326182940100

Offset: 0

Don Knuth, Jul 01 2005

Also the number of non-isomorphic sets of sets with {} that are closed under intersection. Also the number of non-isomorphic set-systems (without {}) covering n + 1 vertices and closed under intersection. - Gus Wiseman, Aug 05 2019

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 28 sets of sets with {} that are closed under intersection:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{1}{2}      {}{123}
             {}{2}{12}     {}{1}{2}
             {}{1}{2}{12}  {}{1}{23}
                           {}{2}{12}
                           {}{3}{123}
                           {}{1}{2}{3}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{3}{23}
                           {}{2}{3}{123}
                           {}{3}{13}{23}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{1}{2}{3}{23}
                           {}{1}{2}{3}{123}
                           {}{2}{3}{13}{23}
                           {}{1}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{1}{2}{3}{13}{23}
                           {}{1}{2}{3}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}
                           {}{1}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

Except a(0) = 1, first differences of A193675.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The labeled version is A102895.
The case also closed under union is A326898.
The covering case is A326883.
Cf. A001930, A102894, A102896, A102897, A193674, A326880, A326881.

a(n > 0) = 2 * A108798(n).

a(6) added (using A193675) by N. J. A. Sloane, Aug 02 2011
Changed a(0) from 2 to 1 by Gus Wiseman, Aug 02 2019
a(7) added (using A108798) by Andrew Howroyd, Aug 10 2019

A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 1, 6, 9, 6, 6, 0, 1, 0, 1, 14, 43, 60, 72, 54, 54, 20, 24, 0, 12, 0, 0, 0, 1, 0, 1, 30, 165, 390, 630, 780, 955, 800, 900, 500, 660, 240, 390, 120, 190, 10, 100, 0, 60, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Aug 01 2019

			Triangle begins:
  1
  0  1
  0  1  2  1
  0  1  6  9  6  6  0  1
  0  1 14 43 60 72 54 54 20 24  0 12  0  0  0  1
Row n = 3 counts the following topologies:
{}{123} {}{1}{123}  {}{1}{12}{123} {}{1}{2}{12}{123}  {}{1}{2}{12}{13}{123}
        {}{2}{123}  {}{1}{13}{123} {}{1}{3}{13}{123}  {}{1}{2}{12}{23}{123}
        {}{3}{123}  {}{1}{23}{123} {}{2}{3}{23}{123}  {}{1}{3}{12}{13}{123}
        {}{12}{123} {}{2}{12}{123} {}{1}{12}{13}{123} {}{1}{3}{13}{23}{123}
        {}{13}{123} {}{2}{13}{123} {}{2}{12}{23}{123} {}{2}{3}{12}{23}{123}
        {}{23}{123} {}{2}{23}{123} {}{3}{13}{23}{123} {}{2}{3}{13}{23}{123}
                    {}{3}{12}{123}
                    {}{3}{13}{123}      {}{1}{2}{3}{12}{13}{23}{123}
                    {}{3}{23}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..254
Wikipedia, Topological space

Row lengths are A000079.
Row sums are A000798.
Cf. A001930, A014466, A102894, A102895, A102896, A102897, A306445, A326876, A326878, A326881.
Columns: A281774 and refs therein.

Table[Length[Select[Subsets[Subsets[Range[n]],{k}],MemberQ[#,{}]&&MemberQ[#,Range[n]]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,4},{k,2^n}]

Terms a(31) and beyond from Andrew Howroyd, Aug 10 2019

A001929 Number of connected topologies on n labeled points.

Original entry on oeis.org

1, 1, 3, 19, 233, 4851, 158175, 7724333, 550898367, 56536880923, 8267519506789, 1709320029453719, 496139872875425839, 200807248677750187825, 112602879608997769049739, 86955243134629606109442219, 91962123875462441868790125305, 132524871920295877733718959290203, 259048612476248175744581063815546423

Offset: 0

N. J. A. Sloane

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

C. M. Bender et al., Combinatorics and Field theory, arXiv:quant-ph/0604164, 2006.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179, Table IV up to 18 points
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
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)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A001928, A001930.

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

A001035 = {1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023};
max = Length[A001035]-1;
B[x_] = Sum[A001035[[k+1]]*x^k/k!, {k, 0, max}];
A[x_] = 1 + Log[B[x]];
A001927 = CoefficientList[ A[x] + O[x]^(max-1), x]*Range[0, max-2]!;
a[n_] := Sum[StirlingS2[n, k] *A001927[[k+1]], {k, 0, n}];
Table[a[n], {n, 0, max -2}] (* Jean-François Alcover, Aug 30 2018, after Vladeta Jovovic *)

a(n) = Sum_{k=0..n} Stirling2(n,k)*A001927(k). - Vladeta Jovovic, Apr 10 2006

More terms from Vladeta Jovovic, Apr 10 2006

a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018

A121337 Number of idempotent relations on n labeled elements.

Original entry on oeis.org

1, 2, 11, 123, 2360, 73023, 3465357

Offset: 0

Florian Kammüller (flokam(AT)cs.tu-berlin.de), Aug 28 2006

A relation r is idempotent if r ; r = r, where ; denotes sequential composition.
From Geoffrey Critzer, Oct 18 2023 : (Start)
a(n) is also the number of maximal subgroups in the semigroup of binary relations on [n].  See Butler and Markowski link.
A binary relation is idempotent iff it is both dense (A355730) and transitive (A006905).
A binary relation is idempotent iff it is both limit dominating (A366194) and limit dominated (A366722).  See Gregory, Kirkland, and Pullman link.
A binary relation R on [n] is idempotent iff the following biconditional statement holds for all x,y in [n]:  There is a cyclic traverse from x to y in G(R) iff (x,y) is in R.  Here, G(R) is the directed graph with self loops allowed (A002416) corresponding to R.  See Rosenblatt link.
Let Q be a quasi-order (A000798) on [n].  Let D(X) be the relation {(x,x):x is in X}.  Let S be a subset of [n] such that: (i) For all x in S, the class in the equivalence relation Q intersect Q^(-1) containing (x,x) is a singleton and (ii) for all x,y in S, the component containing x is not covered by the component containing y in the condensation of G(Q) .  Here, the condensation of G(Q) is the acyclic digraph (A003024) obtained from G(Q) by replacing every strongly connected component (SCC) by a single vertex and all directed edges from one SCC to another with a single directed edge.  Then a relation is idempotent iff it is of the form Q-D(S).  See Schein link.  (End)

			a(2) = 11 because given a set {a,b} of two elements, of the 2^(2*2) = 16 relations on the set, only 5 are not idempotent. - _Michael Somos_, Jul 28 2013
These 5 relations that are not idempotent are: {(a,b)}, {(b,a)}, {(a,b),(b,a)}, {(a,b),(b,a),(b,b)}, {(a,a),(a,b),(b,a)}. - _Geoffrey Critzer_, Aug 07 2016

F. Kammüller, Interactive Theorem Proving in Software Engineering, Habilitationsschrift, Technische Universitaet Berlin (2006).
Ki Hang Kim, Boolean Matrix Theory and Applications, Marcel Decker, 1982.

G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.
D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
F. Kammüller, Counting idempotent relations.
F. Kammüller and J. W. Sanders, Idempotent relations in Isabelle/HOL, International Colloquium on Theoretical Aspects of Computing, ICTAC'04. Volume 3407 of Lecture Notes in Computer Science, Springer-Verlag (2005).
G. Pfeiffer, Counting transitive relations, J. Integer Sequences, Volume 7, 2004, #3.
D. Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research of the National Bureau of Standards, 67B No. 4, 1963.
B. M. Schein, A construction for idempotent binary relations, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.

Cf. A000798 (labeled quasi-orders (or topologies)), A001930 (unlabeled quasi-orders), A001035 (labeled partial orders), A000112 (unlabeled partial orders), A002416, A003024, A366722, A366194, A355730, A006905.

Row sums of A360984.

Prepend[Table[Length[Select[Tuples[Tuples[{0, 1}, n], n], (MatrixPower[#, 2] /. x_ /; x > 0 -> 1) == # &]], {n, 1, 4}], 1] (* Geoffrey Critzer, Aug 07 2016 *)

Offset corrected by James Mitchell, Jul 28 2013

a(1) corrected by Philippe Beaudoin, Aug 11 2015

A000608 Number of connected partially ordered sets with n unlabeled elements.

Original entry on oeis.org

1, 1, 1, 3, 10, 44, 238, 1650, 14512, 163341, 2360719, 43944974, 1055019099, 32664984238, 1303143553205, 66900392672168, 4413439778321689

Offset: 0

N. J. A. Sloane, Simon Plouffe, J. A. Wright

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
G. Melançon, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points [On Brendan McKay's home page]
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
Wikipedia, Illustration n=4, Illustration n=5, Illustration n=6" (2021)
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970. [Annotated scanned copy]
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, Two related abstracts, 1970 and 1972 [Annotated scanned copies]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Inverse Euler transform of A000112.

Cf. A263864 (multiset transform), A342500 (refined by rank).

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[Rest[A000112]] (* Jean-François Alcover, Dec 04 2019, updated Mar 17 2020 *)

More terms from Christian G. Bower, who pointed out connection with A000112, Jan 21 1998 and Dec 12 2001

More terms from Vladeta Jovovic, Jan 04 2006; corrected Jan 15 2006

A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736

Offset: 0

Gus Wiseman, Jul 30 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 6, 32, 418, 23702, 16554476, 1063574497050, 225402367516942398102

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of a set-system that is closed under intersection can be disjoint.

			The a(3) = 32 set-systems:
  {}  {{1}}    {{1}{12}}    {{1}{12}{13}}   {{1}{12}{13}{123}}
      {{2}}    {{1}{13}}    {{2}{12}{23}}   {{2}{12}{23}{123}}
      {{3}}    {{2}{12}}    {{3}{13}{23}}   {{3}{13}{23}{123}}
      {{12}}   {{2}{23}}    {{1}{12}{123}}
      {{13}}   {{3}{13}}    {{1}{13}{123}}
      {{23}}   {{3}{23}}    {{2}{12}{123}}
      {{123}}  {{1}{123}}   {{2}{23}{123}}
               {{2}{123}}   {{3}{13}{123}}
               {{3}{123}}   {{3}{23}{123}}
               {{12}{123}}
               {{13}{123}}
               {{23}{123}}

M. Habib and L. Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.

The case with union instead of intersection is A102896.
The case closed under union and intersection is A326900.
The covering case is A326902.
The connected case is A326903.
The unlabeled version is A326904.
The BII-numbers of these set-systems are A326905.
Cf. A000798, A001930, A006058, A102895, A102898, A182507, A326866, A326876, A326878, A326882.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = 1 + Sum_{k=0, n-1} binomial(n,k)*A102895(k). - Andrew Howroyd, Aug 10 2019

a(5)-a(8) from Andrew Howroyd, Aug 10 2019

A079265 Number of antisymmetric transitive binary relations on n unlabeled points.

Original entry on oeis.org

1, 2, 7, 32, 192, 1490, 15067, 198296, 3398105, 75734592, 2191591226, 82178300654, 3984499220967, 249298391641352, 20089200308020179, 2081351202770089728

Offset: 0

N. J. A. Sloane, Feb 16 2003

Also, number of unconstrained mixed models with n factors.

A. Hess and H. Iyer, Enumeration of mixed linear models and SAS macro for computation of confidence intervals for variance components, presented at Applied Statistics in Agriculture Conference at Kansas State University 2001.

R. Bayon, N. Lygeros and J.-S. Sereni, New progress in enumeration of mixed models, Applied Mathematics E-Notes, 5 (2005), 60-65.
R. Bayon, N. Lygeros and J.-S. Sereni, Nouveaux progrès dans l'énumération des modèles mixtes, in Knowledge discovery and discrete mathematics : JIM'2003, INRIA, Université de Metz, France, 2003, pp. 243-246.
Gunnar Brinkmann and Brendan D. McKay, Counting unlabelled topologies and transitive relations.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
R. Fraïssé and N. Lygeros, Petits posets: dénombrement, représentabilité par cercles et "compenseurs", C. R. Acad. Sci. Paris, Vol. 313, series I, pp. 417-420, 1991.
Ann Marie Hess, Mixed Models Site
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

Cf. A000112 (partial orders), A091073 (transitive relations), A001930 (quasi-orders), A085628 (labeled antisymmetric transitive relations).

Cf. A079263, A006126, A006602, A006896-A006898.

a(10)-a(12) and new description from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(15) from Brinkmann's and McKay's paper by Vladeta Jovovic, Jan 04 2006

A326898 Number of unlabeled topologies with up to n points.

Original entry on oeis.org

1, 2, 5, 14, 47, 186, 904, 5439, 41418, 404501, 5122188, 84623842, 1828876351, 51701216248, 1908493827243, 91755916071736, 5729050033597431

Offset: 0

Gus Wiseman, Aug 02 2019

nonn

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 topologies:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{2}{12}     {}{123}
             {}{1}{2}{12}  {}{2}{12}
                           {}{3}{123}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}

Partial sums of A001930.
The labeled version is A326878.
Cf. A000612, A000798, A003180, A108800, A193675, A306445, A326867, A326882.

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792

Offset: 0

Gus Wiseman, Aug 04 2019

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.

Apart from the offset the same as A193675. - R. J. Mathar, Aug 09 2019

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{2},{1,2}}  {{1,2,3}}
                          {{2},{1,2}}
                          {{3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{3},{1,3},{2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The covering case is A108800(n - 1).
The case with an edge containing all of the vertices is A193674(n - 1).
The case with union instead of intersection is A193674.
The labeled version is A326901.
Cf. A000798, A001930, A006058, A102895, A102898, A326876, A326866, A326878, A326882, A326903, A326906.

a(n > 0) = 2 * A193674(n - 1).

cp's OEIS Frontend

A108800 Number of nonisomorphic systems enumerated by A102895.

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A326882 Irregular triangle read by rows where T(n,k) is the number of finite topologies with n points and k nonempty open sets, 0 <= k <= 2^n - 1.

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A001929 Number of connected topologies on n labeled points.

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A121337 Number of idempotent relations on n labeled elements.

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A000608 Number of connected partially ordered sets with n unlabeled elements.

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A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

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A326901 Number of set-systems (without {}) on n vertices that are closed under intersection.

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A079265 Number of antisymmetric transitive binary relations on n unlabeled points.

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