A326874 - OEIS

%I #9 Jul 31 2019 08:10:01

%S 0,1,2,3,7,8,9,10,11,15,25,27,31,42,43,47,59,63,127,128,129,130,131,

%T 135,136,137,138,139,143,153,155,159,170,171,175,187,191,255,385,387,

%U 391,393,395,399,409,411,415,427,431,443,447,511,642,643,647,650,651,655

%N BII-numbers of abstract simplicial complexes.

%C An abstract simplicial complex is a set of finite nonempty sets (edges) that is closed under taking a nonempty subset of any edge.

%C 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.

%C The enumeration of abstract simplicial complexes by number of covered vertices is given by A307249.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abstract_simplicial_complex">Abstract simplicial complex</a>

%e The sequence of all abstract simplicial complexes together with their BII-numbers begins:

%e     0: {}

%e     1: {{1}}

%e     2: {{2}}

%e     3: {{1},{2}}

%e     7: {{1},{2},{1,2}}

%e     8: {{3}}

%e     9: {{1},{3}}

%e    10: {{2},{3}}

%e    11: {{1},{2},{3}}

%e    15: {{1},{2},{1,2},{3}}

%e    25: {{1},{3},{1,3}}

%e    27: {{1},{2},{3},{1,3}}

%e    31: {{1},{2},{3},{1,2},{1,3}}

%e    42: {{2},{3},{2,3}}

%e    43: {{1},{2},{3},{2,3}}

%e    47: {{1},{2},{3},{1,2},{2,3}}

%e    59: {{1},{2},{3},{1,3},{2,3}}

%e    63: {{1},{2},{3},{1,2},{1,3},{2,3}}

%e   127: {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

%e   128: {{4}}

%e   129: {{1},{4}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[0,100],SubsetQ[bpe/@bpe[#],DeleteCases[Union@@Subsets/@bpe/@bpe[#],{}]]&]

%Y Cf. A006126, A014466, A029931, A048793, A102896, A261005, A307249, A326031, A326872, A326876.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jul 29 2019