

A326943


Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.


9




OFFSET

0,1


COMMENTS

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks 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..7.


FORMULA

Inverse binomial transform of A326945.
a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k).  Andrew Howroyd, Aug 14 2019


EXAMPLE

The a(0) = 2 through a(3) = 6 sets of subsets:
{} {{1}} {{1},{1,2}}
{{}} {{},{1}} {{2},{1,2}}
{{},{1},{2}}
{{},{1},{1,2}}
{{},{2},{1,2}}
{{},{1},{2},{1,2}}


MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]


CROSSREFS

The nonT_0 version is A326906.
The case without empty edges is A309615.
The noncovering version is A326945.
The version not closed under intersection is A326939.
Cf. A003180, A003181, A003465, A059052, A059201, A245567, A316978, A319564, A319637, A326940, A326941, A326942, A326947.
Sequence in context: A156529 A184712 A303225 * A304564 A181265 A093909
Adjacent sequences: A326940 A326941 A326942 * A326944 A326945 A326946


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Aug 08 2019


EXTENSIONS

a(5)a(7) from Andrew Howroyd, Aug 14 2019


STATUS

approved



