A329628 - OEIS

%I #5 Nov 29 2019 01:40:04

%S 0,1,20,52,2880,275520

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

%C 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 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 The sequence of terms together with their corresponding set-systems begins:

%e        0: {}

%e        1: {{1}}

%e       20: {{1,2},{1,3}}

%e       52: {{1,2},{1,3},{2,3}}

%e     2880: {{1,2,3},{1,4},{2,4},{3,4}}

%e   275520: {{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,5}}

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%t First/@GatherBy[Select[Range[0,10000],stableQ[bpe/@bpe[#],SubsetQ[#1,#2]||Intersection[#1,#2]=={}&]&],Length[bpe[#]]&]

%Y The not necessarily intersecting version is A329626.

%Y MM-numbers of intersecting antichains are A329366.

%Y BII-numbers of antichains are A326704.

%Y BII-numbers of intersecting set-systems are A326910.

%Y BII-numbers of intersecting antichains are A329561.

%Y Covering intersecting antichains of sets are A305844.

%Y Non-isomorphic intersecting antichains of multisets are A306007.

%Y Cf. A000120, A048793, A070939, A072639, A316476, A305857, A326031, A326361, A326912, A329560.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Nov 28 2019