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A329628 Smallest BII-number of an intersecting antichain with n edges. 4
0, 1, 20, 52, 2880, 275520 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=0..5.


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}}


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[#]]&]


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




Gus Wiseman, Nov 28 2019



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Last modified January 31 12:13 EST 2023. Contains 359971 sequences. (Running on oeis4.)