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A329628
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Smallest BII-number of an intersecting antichain with n edges.
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4
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OFFSET
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0,3
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COMMENTS
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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.
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.
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LINKS
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Table of n, a(n) for n=0..5.
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EXAMPLE
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The sequence of terms together with their corresponding set-systems begins:
0: {}
1: {{1}}
20: {{1,2},{1,3}}
52: {{1,2},{1,3},{2,3}}
2880: {{1,2,3},{1,4},{2,4},{3,4}}
275520: {{1,2,3},{1,2,4},{1,3,4},{2,3,4},{1,2,5}}
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
First/@GatherBy[Select[Range[0, 10000], stableQ[bpe/@bpe[#], SubsetQ[#1, #2]||Intersection[#1, #2]=={}&]&], Length[bpe[#]]&]
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CROSSREFS
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The not necessarily intersecting version is A329626.
MM-numbers of intersecting antichains are A329366.
BII-numbers of antichains are A326704.
BII-numbers of intersecting set-systems are A326910.
BII-numbers of intersecting antichains are A329561.
Covering intersecting antichains of sets are A305844.
Non-isomorphic intersecting antichains of multisets are A306007.
Cf. A000120, A048793, A070939, A072639, A316476, A305857, A326031, A326361, A326912, A329560.
Sequence in context: A205220 A188250 A329627 * A302885 A059677 A301372
Adjacent sequences: A329625 A329626 A329627 * A329629 A329630 A329631
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KEYWORD
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nonn,more
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AUTHOR
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Gus Wiseman, Nov 28 2019
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STATUS
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approved
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