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 A306445 Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection. 23
 2, 4, 13, 74, 732, 12085, 319988, 13170652, 822378267, 76359798228, 10367879036456, 2029160621690295, 565446501943834078, 221972785233309046708, 121632215040070175606989, 92294021880898055590522262, 96307116899378725213365550192, 137362837456925278519331211455157, 266379254536998812281897840071155592 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS FORMULA 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.) EXAMPLE 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) MATHEMATICA 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 *) PROG (Python 3) 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) CROSSREFS 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. Sequence in context: A132786 A298065 A298714 * A216670 A103845 A055463 Adjacent sequences:  A306442 A306443 A306444 * A306446 A306447 A306448 KEYWORD nonn AUTHOR Yuan Yao, Feb 15 2019 EXTENSIONS a(16)-a(18) from A000798 by Jean-François Alcover, Dec 30 2019 STATUS approved

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Last modified February 26 11:49 EST 2020. Contains 332279 sequences. (Running on oeis4.)