A326361 -id:A326361 - cp's OEIS frontend

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

0, 1, 20, 52, 2880, 275520

Offset: 0

Gus Wiseman, Nov 28 2019

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.

			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}}

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.

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[#]]&]

A326372 Number of intersecting antichains of (possibly empty) subsets of {1..n}.

Original entry on oeis.org

2, 3, 5, 13, 82, 2647, 1422565, 229809982113, 423295099074735261881

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

			The a(0) = 2 through a(3) = 13 antichains:
  {}    {}     {}       {}
  {{}}  {{}}   {{}}     {{}}
        {{1}}  {{1}}    {{1}}
               {{2}}    {{2}}
               {{1,2}}  {{3}}
                        {{1,2}}
                        {{1,3}}
                        {{2,3}}
                        {{1,2,3}}
                        {{1,2},{1,3}}
                        {{1,2},{2,3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}

The case without empty edges is A001206.
The inverse binomial transform is the spanning case A305844.
The unlabeled case is A306007.
Maximal intersecting antichains are A326363.
Intersecting set systems are A051185.
Cf. A000372, A007363, A014466, A305843, A326361, A326362.

a(n) = A001206(n + 1) + 1.

A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 435, 989555, 887050136795, 291072121058024908202443, 14704019422368226413236661148207899662350666147, 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299

Offset: 0

Gus Wiseman, Jul 01 2019

nonn

A set system (set of sets) is intersecting if no two edges are disjoint.

			The a(3) = 3 intersecting set systems with empty intersection:
  {}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

The inverse binomial transform is the covering case A326364.
Set systems with empty intersection are A318129.
Intersecting set systems are A051185.
Intersecting antichains with empty intersection are A326366.
Cf. A000371, A006126, A007363, A014466, A058891, A305844, A307249, A318128, A326361, A326362, A326363, A326365.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],And[#=={}||Intersection@@#=={}]&]],{n,0,4}]

a(n) = A051185(n) - 1 - Sum_{k=1..n-1} binomial(n,k)*A000371(k). - Andrew Howroyd, Aug 12 2019

a(6)-a(9) from Andrew Howroyd, Aug 12 2019

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

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A326372 Number of intersecting antichains of (possibly empty) subsets of {1..n}.

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A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.

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