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Number of collections of subsets of {1, 2, ..., n} that are closed under union and intersection.
31

%I #30 Jan 20 2024 11:28:17

%S 2,4,13,74,732,12085,319988,13170652,822378267,76359798228,

%T 10367879036456,2029160621690295,565446501943834078,

%U 221972785233309046708,121632215040070175606989,92294021880898055590522262,96307116899378725213365550192,137362837456925278519331211455157,266379254536998812281897840071155592

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

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

%F 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.)

%F 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

%e For n = 0, the empty collection and the collection containing the empty set only are both valid.

%e For n = 1, the 2^(2^1)=4 possible collections are also all closed under union and intersection.

%e 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.

%e From _Gus Wiseman_, Jul 31 2019: (Start)

%e The a(0) = 2 through a(4) = 13 sets of sets:

%e {} {} {}

%e {{}} {{}} {{}}

%e {{1}} {{1}}

%e {{},{1}} {{2}}

%e {{1,2}}

%e {{},{1}}

%e {{},{2}}

%e {{},{1,2}}

%e {{1},{1,2}}

%e {{2},{1,2}}

%e {{},{1},{1,2}}

%e {{},{2},{1,2}}

%e {{},{1},{2},{1,2}}

%e (End)

%t Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}] (* _Gus Wiseman_, Jul 31 2019 *)

%t A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]];

%t a[n_] := 1 + Sum[Binomial[n, i]*Binomial[i, i - d]*A000798[[d + 1]], {d, 0, n}, {i, d, n}];

%t a /@ Range[0, Length[A000798] - 1] (* _Jean-François Alcover_, Dec 30 2019 *)

%o (Python)

%o import math

%o # Sequence A000798

%o topo = [1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203]

%o def nCr(n, r):

%o return math.factorial(n) // (math.factorial(r) * math.factorial(n-r))

%o for n in range(len(topo)):

%o ans = 1

%o for d in range(n+1):

%o for i in range(d, n+1):

%o ans += nCr(n,i) * nCr(i, i-d) * topo[d]

%o print(n, ans)

%Y The covering case with {} is A000798.

%Y The case closed under union only is A102897.

%Y The case closed under intersection only is (also) A102897.

%Y The BII-numbers of these set-systems are A326876.

%Y Cf. A001930, A102895, A102896, A326866, A326878, A326882.

%K nonn

%O 0,1

%A _Yuan Yao_, Feb 15 2019

%E a(16)-a(18) from A000798 by _Jean-François Alcover_, Dec 30 2019