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

%I A326750 #13 Jul 15 2024 15:36:30
%S A326750 0,1,2,4,8,16,20,32,36,48,52,64,128,256,260,272,276,292,304,308,320,
%T A326750 512,516,532,544,548,560,564,576,768,772,784,788,800,804,816,820,832,
%U A326750 1024,1040,1056,1072,1088,2048,2064,2068,2080,2084,2096,2100,2112,2304
%N A326750 BII-numbers of clutters (connected antichains of nonempty sets).
%C A326750 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.
%C A326750 Elements of a set-system are sometimes called edges. In an antichain, no edge is a subset or superset of any other edge.
%H A326750 John Tyler Rascoe, <a href="/A326750/b326750.txt">Table of n, a(n) for n = 1..6834</a>
%H A326750 John Tyler Rascoe, <a href="/A326750/a326750.py.txt">Python program</a>.
%F A326750 Intersection of A326749 and A326704.
%e A326750 The sequence of all clutters together with their BII-numbers begins:
%e A326750     0: {}
%e A326750     1: {{1}}
%e A326750     2: {{2}}
%e A326750     4: {{1,2}}
%e A326750     8: {{3}}
%e A326750    16: {{1,3}}
%e A326750    20: {{1,2},{1,3}}
%e A326750    32: {{2,3}}
%e A326750    36: {{1,2},{2,3}}
%e A326750    48: {{1,3},{2,3}}
%e A326750    52: {{1,2},{1,3},{2,3}}
%e A326750    64: {{1,2,3}}
%e A326750   128: {{4}}
%e A326750   256: {{1,4}}
%e A326750   260: {{1,2},{1,4}}
%e A326750   272: {{1,3},{1,4}}
%e A326750   276: {{1,2},{1,3},{1,4}}
%e A326750   292: {{1,2},{2,3},{1,4}}
%e A326750   304: {{1,3},{2,3},{1,4}}
%e A326750   308: {{1,2},{1,3},{2,3},{1,4}}
%e A326750   320: {{1,2,3},{1,4}}
%t A326750 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326750 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]]]]]]]]];
%t A326750 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326750 Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&]
%o A326750 (Python) # see linked program
%Y A326750 The number of clutters spanning n vertices is A048143(n).
%Y A326750 Cf. A000120, A001187, A048793, A070939, A072639, A304986, A326031, A326702, A326753.
%Y A326750 Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
%K A326750 nonn,base
%O A326750 1,3
%A A326750 _Gus Wiseman_, Jul 23 2019