cp's OEIS Frontend

A329627 Smallest BII-number of a clutter (connected antichain) with n edges.

%I A329627 #11 Jan 06 2020 18:09:45
%S A329627 0,1,20,52,308,820,2868,68404,199476,723764
%N A329627 Smallest BII-number of a clutter (connected antichain) with n edges.
%C A329627 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 set-system (finite set of finite nonempty sets of positive integers) 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 A329627 A set-system is an antichain if no edge is a proper subset of any other.
%C A329627 For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.
%e A329627 The sequence of terms together with their corresponding set-systems begins:
%e A329627        0: {}
%e A329627        1: {{1}}
%e A329627       20: {{1,2},{1,3}}
%e A329627       52: {{1,2},{1,3},{2,3}}
%e A329627      308: {{1,2},{1,3},{2,3},{1,4}}
%e A329627      820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
%e A329627     2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
%e A329627    68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}
%e A329627   199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
%e A329627   723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}
%t A329627 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A329627 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A329627 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A329627 First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&],Length[bpe[#]]&]
%Y A329627 The version for MM-numbers is A329555.
%Y A329627 BII-numbers of clutters are A326750.
%Y A329627 Clutters of sets are counted by A048143.
%Y A329627 Minimum BII-numbers of connected set-systems are A329625.
%Y A329627 Minimum BII-numbers of antichains are A329626.
%Y A329627 MM-numbers of connected weak antichains of multisets are A329559.
%Y A329627 Cf. A048793, A070939, A072639, A320275, A322113, A326031, A326704, A326753, A329628, A329632.
%K A329627 nonn,more
%O A329627 0,3
%A A329627 _Gus Wiseman_, Nov 28 2019