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A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection. 5
1, 2, 5, 22, 297, 20536, 16232437, 1063231148918, 225402337742595309857 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

LINKS

Table of n, a(n) for n=0..8.

FORMULA

Binomial transform of A309615.

EXAMPLE

The a(0) = 1 through a(3) = 22 set-systems:

  {}  {}     {}           {}

      {{1}}  {{1}}        {{1}}

             {{2}}        {{2}}

             {{1},{1,2}}  {{3}}

             {{2},{1,2}}  {{1},{1,2}}

                          {{1},{1,3}}

                          {{2},{1,2}}

                          {{2},{2,3}}

                          {{3},{1,3}}

                          {{3},{2,3}}

                          {{1},{1,2},{1,3}}

                          {{2},{1,2},{2,3}}

                          {{3},{1,3},{2,3}}

                          {{1},{1,2},{1,2,3}}

                          {{1},{1,3},{1,2,3}}

                          {{2},{1,2},{1,2,3}}

                          {{2},{2,3},{1,2,3}}

                          {{3},{1,3},{1,2,3}}

                          {{3},{2,3},{1,2,3}}

                          {{1},{1,2},{1,3},{1,2,3}}

                          {{2},{1,2},{2,3},{1,2,3}}

                          {{3},{1,3},{2,3},{1,2,3}}

MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]

CROSSREFS

The covering case is A309615.

T_0 set-systems are A326940.

The version with empty edges allowed is A326945.

Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.

Sequence in context: A068413 A137069 A050994 * A034384 A078419 A241428

Adjacent sequences:  A326956 A326957 A326958 * A326960 A326961 A326962

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 13 2019

EXTENSIONS

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

STATUS

approved

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Last modified January 17 12:48 EST 2020. Contains 330958 sequences. (Running on oeis4.)