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A326950 Number of T_0 antichains of nonempty subsets of {1..n}. 7
1, 2, 4, 12, 107, 6439, 7726965, 2414519001532, 56130437161079183223017 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 A245567, if we assume A245567(0) = 1.

EXAMPLE

The a(0) = 1 through a(3) = 12 antichains:

  {}  {}     {}         {}

      {{1}}  {{1}}      {{1}}

             {{2}}      {{2}}

             {{1},{2}}  {{3}}

                        {{1},{2}}

                        {{1},{3}}

                        {{2},{3}}

                        {{1,2},{1,3}}

                        {{1,2},{2,3}}

                        {{1},{2},{3}}

                        {{1,3},{2,3}}

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

MATHEMATICA

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

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];

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

CROSSREFS

Antichains of nonempty sets are A014466.

T_0 set-systems are A326940.

The covering case is A245567.

Cf. A006126, A059201, A059052, A245567, A319559, A319564, A326030, A326946, A326947.

Sequence in context: A325502 A038791 A327563 * A001696 A276534 A326969

Adjacent sequences:  A326947 A326948 A326949 * A326951 A326952 A326953

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 08 2019

EXTENSIONS

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

STATUS

approved

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Last modified October 15 05:43 EDT 2019. Contains 328026 sequences. (Running on oeis4.)