cp's OEIS Frontend

A329628 Smallest BII-number of an intersecting antichain with n edges.

%I A329628 #5 Nov 29 2019 01:40:04
%S A329628 0,1,20,52,2880,275520
%N A329628 Smallest BII-number of an intersecting antichain with n edges.
%C A329628 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. Elements of a set-system are sometimes called edges.
%C A329628 A set-system is intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.
%e A329628 The sequence of terms together with their corresponding set-systems begins:
%e A329628        0: {}
%e A329628        1: {{1}}
%e A329628       20: {{1,2},{1,3}}
%e A329628       52: {{1,2},{1,3},{2,3}}
%e A329628     2880: {{1,2,3},{1,4},{2,4},{3,4}}
%e A329628   275520: {{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,5}}
%t A329628 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A329628 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A329628 First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#],SubsetQ[#1,#2]||Intersection[#1,#2]=={}&]&],Length[bpe[#]]&]
%Y A329628 The not necessarily intersecting version is A329626.
%Y A329628 MM-numbers of intersecting antichains are A329366.
%Y A329628 BII-numbers of antichains are A326704.
%Y A329628 BII-numbers of intersecting set-systems are A326910.
%Y A329628 BII-numbers of intersecting antichains are A329561.
%Y A329628 Covering intersecting antichains of sets are A305844.
%Y A329628 Non-isomorphic intersecting antichains of multisets are A306007.
%Y A329628 Cf. A000120, A048793, A070939, A072639, A316476, A305857, A326031, A326361, A326912, A329560.
%K A329628 nonn,more
%O A329628 0,3
%A A329628 _Gus Wiseman_, Nov 28 2019