A326750 BII-numbers of clutters (connected antichains of nonempty sets).

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 292, 304, 308, 320, 512, 516, 532, 544, 548, 560, 564, 576, 768, 772, 784, 788, 800, 804, 816, 820, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2068, 2080, 2084, 2096, 2100, 2112, 2304

Offset: 1

Author

Gus Wiseman, Jul 23 2019

Keywords

• nonn
• base

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. 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. In an antichain, no edge is a subset or superset of any other edge.

Examples

			The sequence of all clutters together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..6834
• John Tyler Rascoe, Python program.

Crossrefs

The number of clutters spanning n vertices is A048143(n).

Cf. A000120, A001187, A048793, A070939, A072639, A304986, A326031, A326702, A326753.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&]

Python

# see linked program

Formula

Intersection of A326749 and A326704.