cp's OEIS Frontend

A329626 Smallest BII-number of an antichain with n edges.

%I A329626 #5 Nov 29 2019 01:39:46
%S A329626 0,1,3,11,139,820,2868,35636,199476,723764
%N A329626 Smallest BII-number of an antichain with n edges.
%C A329626 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 A329626 A set-system is an antichain if no edge is a proper subset of any other.
%e A329626 The sequence of terms together with their corresponding set-systems begins:
%e A329626        0: {}
%e A329626        1: {{1}}
%e A329626        3: {{1},{2}}
%e A329626       11: {{1},{2},{3}}
%e A329626      139: {{1},{2},{3},{4}}
%e A329626      820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
%e A329626     2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
%e A329626    35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
%e A329626   199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
%e A329626   723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}
%t A329626 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A329626 stableQ[u_]:=!Apply[Or,Outer[#1=!=#2&&SubsetQ[#1,#2]&,u,u,1],{0,1}];
%t A329626 First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#]]&],Length[bpe[#]]&]
%Y A329626 The connected case is A329627.
%Y A329626 The intersecting case is A329628.
%Y A329626 BII-numbers of antichains are A326704.
%Y A329626 Antichain covers are A006126.
%Y A329626 Cf. A048143, A048793, A070939, A303362, A319837, A326031, A326750, A329555, A329560, A329561, A329625.
%K A329626 nonn,more
%O A329626 0,3
%A A329626 _Gus Wiseman_, Nov 28 2019