

A329628


Smallest BIInumber of an intersecting antichain with n edges.


4




OFFSET

0,3


COMMENTS

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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets of positive integers) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges. Elements of a setsystem are sometimes called edges.
A setsystem is intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.


LINKS

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


EXAMPLE

The sequence of terms together with their corresponding setsystems 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}}


MATHEMATICA

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[#]]&]


CROSSREFS

The not necessarily intersecting version is A329626.
MMnumbers of intersecting antichains are A329366.
BIInumbers of antichains are A326704.
BIInumbers of intersecting setsystems are A326910.
BIInumbers of intersecting antichains are A329561.
Covering intersecting antichains of sets are A305844.
Nonisomorphic intersecting antichains of multisets are A306007.
Cf. A000120, A048793, A070939, A072639, A316476, A305857, A326031, A326361, A326912, A329560.
Sequence in context: A205220 A188250 A329627 * A008524 A302885 A059677
Adjacent sequences: A329625 A329626 A329627 * A329629 A329630 A329631


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Nov 28 2019


STATUS

approved



