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

0, 1, 20, 52, 2880, 275520

Offset: 0

Author

Gus Wiseman, Nov 28 2019

Keywords

• nonn
• more

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 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.

Examples

			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}}
		

Crossrefs

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.

Programs

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[#]]&]