|
%I #8 Aug 12 2019 22:32:31
%S 1,2,4,20,1232
%N Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.
%C 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}}. An antichain is a set of sets, none of which is a subset of any other.
%e Non-isomorphic representatives of the a(0) = 1 through a(3) = 20 set-systems:
%e {} {} {} {}
%e {{1}} {{1}} {{1}}
%e {{1},{2}} {{1},{2}}
%e {{1},{2},{1,2}} {{1},{2},{3}}
%e {{1},{2},{1,2}}
%e {{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{2,3}}
%e {{1},{2},{1,3},{2,3}}
%e {{1},{2},{3},{1,2,3}}
%e {{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{1,3},{2,3}}
%e {{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{2,3},{1,2,3}}
%e {{2},{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3}}
%e {{3},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,3},{2,3},{1,2,3}}
%e {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%Y Unlabeled set-systems are A000612.
%Y Unlabeled set-systems whose dual is strict are A326946.
%Y The version with empty edges allowed is A326951.
%Y The labeled version is A326965.
%Y The version where the dual is not required to be strict is A326971.
%Y The covering version is A326974 (first differences).
%Y Cf. A059523, A319559, A319637, A326973, A326976, A326977, A326979.
%K nonn,more
%O 0,2
%A _Gus Wiseman_, Aug 11 2019
|
