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A309615 Number of T_0 set-systems covering n vertices that are closed under intersection. 5
1, 1, 2, 12, 232, 19230, 16113300, 1063117943398, 225402329237199496416 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

First differs from A182507 at a(5) = 19230, A182507(5) = 12848.

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

a(n) = A326943(n) - A326944(n).

a(n) = Sum_{k = 1..n} s(n,k) * A326901(k - 1) where s = A048994.

a(n) = Sum_{k = 1..n} s(n,k) * A326902(k) where s = A048994.

EXAMPLE

The a(0) = 1 through a(3) = 12 set-systems:

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

             {{2},{1,2}}  {{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}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]

CROSSREFS

The version with empty edges allowed is A326943.

Cf. A003465, A059052, A059201, A319637, A326880, A326881, A326901, A326902, A326905, A326944, A326945.

Sequence in context: A009359 A011807 A182507 * A166316 A011840 A296462

Adjacent sequences:  A309612 A309613 A309614 * A309616 A309617 A309618

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 11 2019

STATUS

approved

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Last modified February 19 01:03 EST 2020. Contains 332028 sequences. (Running on oeis4.)