A001206 -id:A001206 - cp's OEIS frontend

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 15, 30, 30, 5, 1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1, 1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 28 2000

An antichain is called intersecting (or proper) antichain if every two members have a nonempty intersection. Row sums give the number of intersecting antichains on a labeled n-set or n-variable Boolean functions in the Post class F(7,2) or self-dual monotone Boolean functions of n+1 variables. Cf. A001206.

			1;
1, 1;
1, 3;
1, 7, 3, 1;
1, 15, 30, 30, 5;
1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1;
1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7;

Jovovic V., Kilibarda G., The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
Pogosyan G., Miyakawa M., Nozaki A., Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Cf. A001206, A032263, A051303-A051307, A036239, A051180-A051185, A016269, A047707, A051112-A051118, A000372.

T(n, 0)=1, T(n, 1)=2^n-1, T(n, 2)=A032263(n), T(n, 3)=A051303(n), T(n, 4)=A051304(n), T(n, 5)=A051305(n), T(n, 6)=A051306(n), T(n, 7)=A051307(n).

A107766 Number of self-dual monotonic Boolean functions on n variables (where the result depends on all n variables).

Original entry on oeis.org

1, 0, 1, 4, 46, 2284, 1405428, 229798671816, 423295097006496421879

Offset: 1

Don Knuth, Jun 11 2005

Hans-J. Bandelt and Jarmila Hedlíková, Median algebras, Discrete Mathematics, 45 (1983), 1-30.

Cf. A001206, A107765.

A001206(n) = Sum_{k=1..n} binomial(n,k)*a(k).

a(7)-a(8) from Vladeta Jovovic, Jun 13 2005

a(9) (using formula) from Pontus von Brömssen, Dec 29 2023

A326375 Number of intersecting antichains of subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

2, 2, 2, 3, 29, 1961, 1379274, 229755337550, 423295079757497714060

Offset: 0

Gus Wiseman, Jul 03 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(4) = 29 antichains:
  {}
  {{}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

The case without empty edges is A326366.
Intersecting antichains are A326372.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
Cf. A001206, A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326363, A326365, A326373.

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]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

a(n) = A326366(n) + 1.

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A088895 Number of intersecting T_0-antichains on a labeled n-set.

Original entry on oeis.org

1, 2, 2, 4, 41, 2104, 1387915, 229780525655

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Nov 28 2003

An antichain on a set is a T_0-antichain if for every two distinct pointsf the set there exists a member of the antichain containing one but not the other point.

Cf. A001206, A051185, A059083.

A248929 Triangle read by rows: T(n,k) = PIP(n,k) is the number of maximal families of sets from {1,2,...,n} with the property that if A and B are sets in the family, then |AB|>=k.

Original entry on oeis.org

1, 2, 1, 4, 3, 1, 12, 7, 4, 1, 81, 25, 11, 5, 1, 2646, 216, 46, 16, 6, 1, 1422564, 12784, 477, 77, 22, 7, 1, 229809982112, 11115851, 45104, 925, 120, 29, 8, 1

Offset: 1

John M. Ingram, Oct 17 2014

A family of sets has the k (k>=1) pairwise intersection property (PIPk) means that if A and B are sets in the family, then |AB|>=k. A family of sets with PIPk is maximal means no set can be added to the family while maintaining PIPk. (If C is a set not in the family, then there exists a set D in the family such that |CD|<=k-1.) PIP(n,k) is the number of maximal families of sets from {1,2,...,n} with PIPk.

			Triangle PIP(n,k) begins:
n\k 1             2         3      4       5       6     7...
1   1
2   2             1
3   4             3         1
4   12            7         4      1
5   81            25        11     5       1
6   2646          216       46     16      6       1
7   1422564       12784     477    77      22      7     1
8   229809982112  11115851  45104  925     120     29    8
9                                  129315  1633    177   37
10                                         320026  2686  250
11                                                       4181

Ian Anderson, Combinatorics of Finite Sets, Oxford University Press, 1987, pages 1-9.

Cf. A001206 (first column).

PIP(k,k)=1
PIP(k+1,k)=C(k+1,1)=k+1
PIP(k+2,k)=C(k+2,2)+1
PIP(k+3,k)=2*C(k+3,3)+C(k+3,1)
PIP(k+4,k)=12*C(k+4,4)+C(k+4,3)+C(k+4,2)+1

Term PIP(6,2) (12778 should be 12784) in the data sequence and in the example table corrected by John M. Ingram, Nov 02 2014
Another row added to the data sequence by John M. Ingram, Nov 02 2014
Several new terms added to the example table by John M. Ingram, Nov 02 2014

cp's OEIS Frontend

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

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A107766 Number of self-dual monotonic Boolean functions on n variables (where the result depends on all n variables).

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A326375 Number of intersecting antichains of subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

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A088895 Number of intersecting T_0-antichains on a labeled n-set.

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A248929 Triangle read by rows: T(n,k) = PIP(n,k) is the number of maximal families of sets from {1,2,...,n} with the property that if A and B are sets in the family, then |AB|>=k.

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