0,1

Table of n, a(n) for n=0..18.

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.)

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)

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 *)

(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)

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

nonn

Yuan Yao, Feb 15 2019

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

approved