A326874 BII-numbers of abstract simplicial complexes.

0, 1, 2, 3, 7, 8, 9, 10, 11, 15, 25, 27, 31, 42, 43, 47, 59, 63, 127, 128, 129, 130, 131, 135, 136, 137, 138, 139, 143, 153, 155, 159, 170, 171, 175, 187, 191, 255, 385, 387, 391, 393, 395, 399, 409, 411, 415, 427, 431, 443, 447, 511, 642, 643, 647, 650, 651, 655

Offset: 1

Author

Gus Wiseman, Jul 29 2019

Keywords

• nonn

Comments

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

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

Examples

			The sequence of all abstract simplicial complexes together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    7: {{1},{2},{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   15: {{1},{2},{1,2},{3}}
   25: {{1},{3},{1,3}}
   27: {{1},{2},{3},{1,3}}
   31: {{1},{2},{3},{1,2},{1,3}}
   42: {{2},{3},{2,3}}
   43: {{1},{2},{3},{2,3}}
   47: {{1},{2},{3},{1,2},{2,3}}
   59: {{1},{2},{3},{1,3},{2,3}}
   63: {{1},{2},{3},{1,2},{1,3},{2,3}}
  127: {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
  128: {{4}}
  129: {{1},{4}}
		

Links

• Wikipedia, Abstract simplicial complex

Crossrefs

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

Programs

Mathematica

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