A326880 -id:A326880 - cp's OEIS frontend

A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

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

Offset: 0

Gus Wiseman, Aug 03 2019

Differs from A108800 in having a(0) = 2 instead of 1.

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 28 sets of sets:
  {}    {{1}}    {{12}}          {{123}}
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{2}{12}}       {{3}{123}}
                 {{}{2}{12}}     {{23}{123}}
                 {{1}{2}{12}}    {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{23}{123}}
                                 {{1}{23}{123}}
                                 {{3}{23}{123}}
                                 {{13}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{13}{23}{123}}
                                 {{2}{3}{23}{123}}
                                 {{2}{13}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{12}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{2}{13}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{12}{13}{23}{123}}
                                 {{2}{3}{13}{23}{123}}
                                 {{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{3}{12}{13}{23}{123}}
                                 {{2}{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{12}{13}{23}{123}}
                                 {{1}{2}{3}{12}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case without empty sets is A108798.
The case with a single covering edge is A108800.
First differences of A193675.
The case also closed under intersection is A326898 for n > 0.
The labeled version is A326906.
The same for union instead of intersection is (also) A326907.
Cf. A001930, A102895, A108798, A193674, A193675, A326880, A326881, A326883, A326898, A326908.

a(7) added from A108800 by Andrew Howroyd, Aug 10 2019

A326913 BII-numbers of set-systems (without {}) closed under union and intersection.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 384, 512, 514, 640, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1285, 1408, 1536, 1538, 1542, 1664, 1920, 2048, 2056, 2176

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

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.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all set-systems closed under union and intersection together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  72: {{3},{1,2,3}}

Cf. A048793, A102894, A102895, A102896, A102897, A326031, A326875, A326876, A326878, A326880, A326901.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Union@@@Tuples[bpe/@bpe[#],2]]&&SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]

cp's OEIS Frontend

A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

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