A326882 -id:A326882 - cp's OEIS frontend

A000798 Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.

1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203

Offset: 0

N. J. A. Sloane

From Altug Alkan, Dec 18 2015 and Feb 28 2017: (Start)
a(p^k) == k+1 (mod p) for all primes p. This is proved by Kizmaz at On The Number Of Topologies On A Finite Set link. For proof see Theorem 2.4 in page 2 and 3. So a(19) == 2 (mod 19).
a(p+n) == A265042(n) (mod p) for all primes p. This is also proved by Kizmaz at related link, see Theorem 2.7 in page 4. If n=2 and p=17, a(17+2) == A265042(2) (mod 17), that is a(19) == 51 (mod 17). So a(19) is divisible by 17.
In conclusion, a(19) is a number of the form 323*n - 17. (End)
The BII-numbers of finite topologies without their empty set are given by A326876. - Gus Wiseman, Aug 01 2019
From Tian Vlasic, Feb 23 2022: (Start)
Although no general formula is known for a(n), by considering the number of topologies with a fixed number of open sets, it is possible to explicitly represent the sequence in terms of Stirling numbers of the second kind.
For example: a(n,3) = 2*S(n,2), a(n,4) = S(n,2) + 6*S(n,3), a(n,5) = 6*S(n,3) + 24*S(n,4).
Lower and upper bounds are known: 2^n <= a(n) <= 2^(n*(n-1)), n > 1.
This follows from the fact that there are 2^(n*(n-1)) reflexive relations on a set with n elements.
Furthermore: a(n+1) <= a(n)*(3a(n)+1). (End)

			From _Gus Wiseman_, Aug 01 2019: (Start)
The a(3) = 29 topologies are the following (empty sets not shown):
  {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}
(End)

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.
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 229.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 243.
Levinson, H.; Silverman, R. Topologies on finite sets. II. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 699--712, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561090 (81c:54006)
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).
For further references concerning the enumeration of topologies and posets see under A001035.
G.H. Patil and M.S. Chaudhary, A recursive determination of topologies on finite sets, Indian Journal of Pure and Applied Mathematics, 26, No. 2 (1995), 143-148.

V. I. Arnautov and A. V. Kochina, Method for constructing one-point expansions of a topology on a finite set and its applications, Bul. Acad. Stiinte Republ. Moldav. Matem. 3 (64) (2010) 67-76.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Moussa Benoumhani and Ali Jaballah, Chains in lattices of mappings and finite fuzzy topological spaces, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 99-111.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5.
Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 points.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table IV).
J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39.
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]
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966 [Annotated scanned copy]
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date [Annotated scanned copy]
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, The number of partially ordered (labeled) sets, Preprint, 1989. (Annotated scanned copy)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313. [Annotated scanned copy]
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313.
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
Loic Foissy, Claudia Malvenuto, and Frederic Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014.
Loic Foissy, Claudia Malvenuto, and Frederic Patras, Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. .
L. Foissy and C. Malvenuto, The Hopf algebra of finite topologies and T-partitions, arXiv preprint arXiv:1407.0476 [math.RA], 2014.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
S. Giraudo, J.-G. Luque, L. Mignot and F. Nicart, Operads, quasiorders and regular languages, arXiv preprint arXiv:1401.2010 [cs.FL], 2014.
D. J. Greenhoe, Properties of distance spaces with power triangle inequalities, ResearchGate, 2015.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550[math.CO], 2012.
M. Y. Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015-2019.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. Amer. Math. Soc., 25 (1970), 276-282.
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
Messaoud Kolli, On the cardinality of the T_0-topologies on a finite set, International Journal of Combinatorics, Volume 2014 (2014), Article ID 798074, 7 pages.
Sami Lazaar, Houssem Sabri, and Randa Tahri, Structural and Numerical Studies of Some Topological Properties for Alexandroff Spaces, Bull. Iran. Math. Soc. (2021).
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
M. Rayburn, On the Borel fields of a finite set, Proc. Amer. Math.. Soc., 19 (1968), 885-889. [Annotated scanned copy]
M. Rayburn and N. J. A. Sloane, Correspondence, 1974
D. Rusin, More info and references [Broken link]
D. Rusin, More info and references [Cached copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198. [Annotated scanned copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Swartz and Nicholas J. Werner, Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau, arXiv:1709.05390 [math.CO], 2017.
Ivo Terek, Smooth Manifolds, Williams College (2025). See p. 6.
Wietske Visser, Koen V. Hindriks and Catholijn M. Jonker, Goal-based Qualitative Preference Systems, 2012.
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
J. A. Wright, Letter to N. J. A. Sloane, Nov 21 1970, with four enclosures
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
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 "core" sequences

Row sums of A326882.
Cf. A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Cf. A102894, A102895, A102897, A306445, A326866, A326876, A326878, A326881.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],DeleteCases[Intersection@@@Tuples[#,2],{}]]]&]],{n,0,3}] (* Gus Wiseman, Aug 01 2019 *)

a(n) = Sum_{k=0..n} Stirling2(n, k)*A001035(k).
E.g.f.: A(exp(x) - 1) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014
It is known that log_2(a(n)) ~ n^2/4. - Tian Vlasic, Feb 23 2022

Two more terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

a(17)-a(18) are from Brinkmann's and McKay's paper. - Vladeta Jovovic, Jun 10 2007

A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 9, 33, 139, 718, 4535, 35979, 363083, 4717687, 79501654, 1744252509, 49872339897, 1856792610995, 89847422244493, 5637294117525695

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 topologies:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{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}
(End)

Loic Foissy, Claudia Malvenuto, Frederic Patras, Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. .
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (but the last entry is wrong).
M. Kolli, On the cardinality of the T_0-topologies on a finite set, Preprint, 2014.
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).
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, personal communication.
For further references concerning the enumeration of topologies and posets see under A000112 and A001035.

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5
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.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
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.
S. R. Finch, Transitive relations, topologies and partial orders
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
L. Foissy, C. Malvenuto, and F. Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014-2015.
Misha Gavrilovich and Misha Rabinovich, The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms, 2024. See p. 11.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550, 2012. - From _N. J. A. Sloane_, Nov 09 2012
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
D. Rusin, Further information and references [Broken link]
D. Rusin, Further information and references [Cached 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, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Swartz and Nicholas J. Werner, Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau, arXiv:1709.05390 [math.CO], 2017.
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Digraph Topology.
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Wikipedia Topological space
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. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057, A001928, A001929.
The case with unions only is A108798.
The case with intersections only is (also) A108798.
Partial sums are A326898 (the non-covering case).
Cf. A000612, A003180, A108800, A193674, A306445, A326876, A326878, A326882.

a(8)-a(12) from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(16) from Brinkmann's and McKay's paper, sent by Vladeta Jovovic, Jan 04 2006

A306445 Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.

Original entry on oeis.org

2, 4, 13, 74, 732, 12085, 319988, 13170652, 822378267, 76359798228, 10367879036456, 2029160621690295, 565446501943834078, 221972785233309046708, 121632215040070175606989, 92294021880898055590522262, 96307116899378725213365550192, 137362837456925278519331211455157, 266379254536998812281897840071155592

Offset: 0

Yuan Yao, Feb 15 2019

nonn

			For n = 0, the empty collection and the collection containing the empty set only are both valid.
For n = 1, the 2^(2^1)=4 possible collections are also all closed under union and intersection.
For n = 2, there are only 3 invalid collections, namely the collections containing both {1} and {2} but not both {1,2} and the empty set. Hence there are 2^(2^2)-3 = 13 valid collections.
From _Gus Wiseman_, Jul 31 2019: (Start)
The a(0) = 2 through a(4) = 13 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}
(End)

R. Stanley, Enumerative Combinatorics, volume 1, second edition, Exercise 3.46.

The covering case with {} is A000798.
The case closed under union only is A102897.
The case closed under intersection only is (also) A102897.
The BII-numbers of these set-systems are A326876.
Cf. A001930, A102895, A102896, A326866, A326878, A326882.

Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}] (* Gus Wiseman, Jul 31 2019 *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
a[n_] := 1 + Sum[Binomial[n, i]*Binomial[i, i - d]*A000798[[d + 1]], {d, 0, n}, {i, d, n}];
a /@ Range[0, Length[A000798] - 1] (* Jean-François Alcover, Dec 30 2019 *)

import math
# Sequence A000798
topo = [1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203]
def nCr(n, r):
    return math.factorial(n) // (math.factorial(r) * math.factorial(n-r))
for n in range(len(topo)):
    ans = 1
    for d in range(n+1):
        for i in range(d, n+1):
            ans += nCr(n,i) * nCr(i, i-d) * topo[d]
    print(n, ans)

a(n) = 1 + Sum_{d=0..n} Sum_{i=d..n} C(n,i)*C(i,i-d)*A000798(d). (Follows by caseworking on the maximal and minimal set in the collection.)

E.g.f.: exp(x) + exp(x)^2*B(exp(x)-1) where B(x) is the e.g.f. for A001035 (after Stanley reference above). - Geoffrey Critzer, Jan 19 2024

a(16)-a(18) from A000798 by Jean-François Alcover, Dec 30 2019

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

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

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).

A326900 Number of set-systems on n vertices that are closed under union and intersection.

Original entry on oeis.org

1, 2, 6, 29, 232, 3032, 62837, 2009408, 97034882, 6952703663, 728107141058, 109978369078580, 23682049666957359, 7195441649260733390, 3056891748255795885338, 1801430622263459795017565, 1462231768717868324127642932, 1624751185398704445629757084188, 2457871026957756859612862822442301

Offset: 0

Gus Wiseman, Aug 04 2019

nonn

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

			The a(0) = 1 through a(3) = 29 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1,2}}      {{3}}
             {{1},{1,2}}  {{1,2}}
             {{2},{1,2}}  {{1,3}}
                          {{2,3}}
                          {{1,2,3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{1,2,3}}
                          {{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}}

Binomial transform of A006058 (the covering case).
The case closed under union only is A102896.
The case with {} allowed is A306445.
The BII-numbers of these set-systems are A326876.
The case closed under intersection only is A326901.
The unlabeled version is A326908.
Cf. A000798, A001930, A102895, A102898, A326866, A326878, A326882.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}]
(* Second program: *)
A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {, }][[All, 2]];
a[n_] := Sum[Binomial[n, k] A006058[[k + 1]], {k, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, Jan 01 2020 *)

a(16)-a(18) from A006058 by Jean-François Alcover, Jan 01 2020

A326902 Number of set-systems (without {}) covering n vertices that are closed under intersection.

Original entry on oeis.org

1, 1, 3, 19, 319, 21881, 16417973, 1063459099837, 225402359008808647339

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(0) = 1 through a(3) = 19 set-systems:
  {}  {{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},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{3},{1,3},{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}}

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A108800(n - 1).
The non-covering case is A326901.
The connected case is A326903.
Cf. A000798, A001930, A102895, A102898, A182507, A326866, A326876, A326878, A326882, A326900, A326901, A326904.

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

Inverse binomial transform of A326901. - Andrew Howroyd, Aug 10 2019

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

A326903 Number of set-systems (without {}) on n vertices that are closed under intersection and have an edge containing all of the vertices, or Moore families without {}.

Original entry on oeis.org

0, 1, 3, 16, 209, 11851, 8277238, 531787248525, 112701183758471199051

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.

If {} is allowed, we get Moore families (A102896, cf A102895).

			The a(1) = 1 through a(3) = 16 set-systems:
  {{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}}

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

The case closed under union and intersection is A006058.
The case with union instead of intersection is A102894.
The unlabeled version is A193674.
The case without requiring the maximum edge is A326901.
The covering case is A326902.
Cf. A000798, A001930, A102895, A102898, A326866, A326876, A326878, A326882, A326904.

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

a(n) = A326901(n) / 2 for n > 0. - Andrew Howroyd, Aug 10 2019

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

cp's OEIS Frontend

A000798 Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.

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A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

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A306445 Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.

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A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

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

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A326898 Number of unlabeled topologies with up to n points.

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

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A326900 Number of set-systems on n vertices that are closed under union and intersection.

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